user11541: Consider the fact that our Nirvana depends on one crucial truth and that is law of Karma. Good karma bear good fruit and bad karma generates bad results. 
Why didn't Buddha say that Fifth Noble Truth is law of Karma ?

user471: Kamma is already covered in the first noble truth that there is suffering.
From AN 5.57:

“This noble disciple reflects thus: ‘I am not the only one who is the
  owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s
  origin, kamma as one’s relative, kamma as one’s resort; who will be
  the heir of whatever kamma, good or bad, that one does. All beings
  that come and go, that pass away and undergo rebirth, are owners of
  their kamma, heirs of their kamma; all have kamma as their origin,
  kamma as their relative, kamma as their resort; all will be heirs of
  whatever kamma, good or bad, that they do.’ As he often reflects on
  this theme, the path is generated. He pursues this path, develops it,
  and cultivates it. As he does so, the fetters are entirely abandoned
  and the underlying tendencies are uprooted.

The noble disciple reflects that he is not the only one. He reflects that all beings are subject to kamma, which results in their birth, death and rebirth based on their kamma. This is one aspect of suffering. Reflecting thus, he chooses to follow the path laid out in the fourth noble truth, in order to become free from suffering.
But at the same time, we should not adopt the Jain fallacy that everything that happens is due to kamma, as seen in the Sivaka Sutta:

"There are cases where some feelings arise based on phlegm... based on
  internal winds... based on a combination of bodily humors... from the
  change of the seasons... from uneven care of the body... from harsh
  treatment... from the result of kamma. You yourself should know how
  some feelings arise from the result of kamma. Even the world is agreed
  on how some feelings arise from the result of kamma. So any brahmans &
  contemplatives who are of the doctrine & view that whatever an
  individual feels — pleasure, pain, neither pleasure-nor-pain — is
  entirely caused by what was done before — slip past what they
  themselves know, slip past what is agreed on by the world. Therefore I
  say that those brahmans & contemplatives are wrong."


user13806: Good karma gives you good destination, and bad gives you bad destination. But it doesn't matter good or bad. All birth is suffering: because where ever born then there is aging and dying. So karma is in the second noble truth:

yāyaṃ taṇhā ponobbhavikā

ponobbhavikā ("leading to rebirth") is new karma. 

user43: MN 19 Dvedhavitakka Sutta:

Before my self-awakening, back when I was still just an unawakened Bodhisatta, the thought occurred to me: What if I divide all my thoughts into two groups?' So I made thinking imbued with sensuality, negativity, harmfulness etc. one group, and thinking imbued with renunciation, non-negativity, & harmlessness etc. - another group.

So yes, if you understand karma as dividing all your action (physical, verbal, and mental) into wholesome and unwholesome, and then minimizing one and maximizing the other - then yes, this is a key teaching. And in a way, it's a complete teaching - one could follow it without asking further questions, and by figuring out all implications arrive at Buddha's result.
However, an explanation like "To avoid all evil, to cultivate good, and to cleanse one's mind — this is the teaching of the Buddhas." (Dhammapada) lacks context: WHY? Why do good? What's wrong with evil? What will I get if I follow this teaching?
So Four Noble Truths structure the teaching like a doctor would structure a consultation with the patient. First it discusses the symptoms, then the diagnosis (the root cause of the symptoms), then it states that removing the root cause will fix the problem, then it offers a methodical plan of healing.
The fourth truth, the Eighfold Path, is 100% based on the "two kinds of thoughts" and the idea of good vs. bad karma. Every item in Eightfold Path starts with the word Samma, which means "correctly done, good quality". So every item explains right and wrong, not just applied to Physical, Verbal, and Mental action - but also applied to View, Livelihood, Practice, Mindfulness, Meditation and so on.
Finally, as Buddha explained in MN 101, it is wrong view to assume that liberation happens as a result of good karma. Instead, according to Buddha's teaching in MN 101, liberation from dukkha is achieved by "abandoning of unskillful mental qualities and the attainment of skillful mental qualities in the here-&-now": 

When I exert a [physical, verbal, or mental] fabrication against this cause of stress, then from the fabrication of exertion there is dispassion. When I look on with equanimity at that cause of stress, then from the development of equanimity there is dispassion.' So he exerts a fabrication against the cause of stress where there comes dispassion from the fabrication of exertion, and develops equanimity with regard to the cause of stress where there comes dispassion from the development of equanimity. Thus the stress coming from the cause of stress for which there is dispassion through the fabrication of exertion is exhausted & the stress resulting from the cause of stress for which there is dispassion through the development of equanimity is exhausted ...  
On seeing a form with the eye, he does not grasp at any theme or details by which — if he were to dwell without restraint over the faculty of the eye — evil, unskillful qualities such as greed or distress might assail him. On hearing a sound with the ear... On smelling an odor with the nose... On tasting a flavor with the tongue... On touching a tactile sensation with the body... On cognizing an idea with the intellect, he does not grasp at any theme or details by which — if he were to dwell without restraint over the faculty of the intellect — evil, unskillful qualities such as greed or distress might assail him. Endowed with this noble restraint over the sense faculties, he is inwardly sensitive to the pleasure of being blameless. ...  
Abandoning covetousness with regard to the world, he dwells with an awareness devoid of covetousness. He cleanses his mind of covetousness. Abandoning ill will and anger, he dwells with an awareness devoid of ill will, sympathetic with the welfare of all living beings. He cleanses his mind of ill will and anger. Abandoning sloth and drowsiness, he dwells with an awareness devoid of sloth and drowsiness, mindful, alert, percipient of light. He cleanses his mind of sloth and drowsiness. Abandoning restlessness and anxiety, he dwells undisturbed, his mind inwardly stilled. He cleanses his mind of restlessness and anxiety. Abandoning uncertainty, he dwells having crossed over uncertainty, with no perplexity with regard to skillful mental qualities. He cleanses his mind of uncertainty. ...  
Having abandoned these five hindrances — imperfections of awareness that weaken discernment — then, quite withdrawn from sensual pleasures, withdrawn from unskillful mental qualities, he enters and remains in the first jhana ... second jhana ... third jhana ... fourth jhana ...  
With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, the monk directs and inclines it to the knowledge of the ending of the mental fermentations. He discerns, as it has come to be, that ... These are mental fermentations... This is the origination of fermentations... This is the cessation of fermentations... This is the way leading to the cessation of fermentations.' His heart, thus knowing, thus seeing, is released from the fermentation of sensuality, the fermentation of becoming, the fermentation of ignorance. With release, there is the knowledge, 'Released.' He discerns that 'Birth is ended, the holy life fulfilled, the task done. There is nothing further for this world. ...  
Such is the teaching of the Tathagata.


user111: Nibbana does not depend on anything. That's why it is called 'uncaused'.
Karma is covered in the Second Noble Truth. Craving for sensual pleasures, craving for becoming and craving for non-becoming are all Karma.

user11237: The whole structure of the four truths can be seen as a teaching about karma (cause and effect); that's why the four truths are sometimes called the "two sets of cause and effect"; see, for example:
The third paragraph in The Four Noble Truths by Theravadin teacher Thanissaro Bhikkhu:

To counter this problem, the four noble truths drop ideas of me/not
  me, and being/not being, and replace them with two sets of variables:
  cause and effect, skillful and unskillful. In other words, there is
  the truth of stress and suffering (unskillful effect), the truth of
  the origination of stress (unskillful cause), the truth of the
  cessation of stress (skillful effect), and the truth of the path to
  the cessation of stress (skillful cause).

The section titled "The Two Sets of Cause and Effect" in The Four Noble Truths by Tibetan teacher Geshe Tashi Tsering:

One of the key beliefs in Buddhism is that nothing comes into being
  without a cause, and this law of cause and effect is a fundamental
  point for understanding the four noble truths. The following table
  shows two sets of cause and effect: suffering as the result of the
  origin of suffering and cessation as the result of the path.
               Result            Cause

Problem        1. Suffering      2. Origin

Solution       3. Cessation      4. Path

Although these two sets of cause and effect are very specific, they
  follow the rule of all causes and effects, and in particular the type
  of cause and effect called karma in Buddhism.


user254: Other answers say that kamma is already covered by, or implicit in, the first and/or second noble truths.
I think it's related to the fourth noble truth -- i.e. you say that kamma is important to attaining nirvana (and I think that "the path towards attaining nirvana" is what the fourth noble truth is about).
Or, perhaps, you can see "bad karma" and "good karma" in the second and third noble truths respectively.

I find the logical structure of the four noble truths satisfying or persuasive.
The first three have the structure of a syllogism:

premise: dukkha exists, and is this and that
premise: dukkha is associated with craving
theorem: dukkha ceases when craving ceases

So that's the doctrine "in a nutshell".
The fourth is, then, an introduction to the rest of Buddhism:

to begin to implement that in practice, see the noble eightfold way


I think that (as well as karma) there are other important aspects of dharma -- e.g. anatta, dependent origination, the sangha, the qualities conducive to Enlightenment, and so on.
So I don't see why you see the need to "privilege" karma (as more important than other doctrines) -- but it's helpful that there is a doctrine with only 4 (memorable) truths, rather than (less memorable) 20. The "4 truths" served as (and still serve as) an introduction to the dharma.

Or perhaps you're saying that the doctrine of karma should be privileged because it's the single most important doctrine to remember. If so, maybe you're not wrong; but there are also some other dharmas that people have been inclined to say are the most important -- for example, "right view"; or maybe "harmlessness", "generosity", "skilful virtue", "admirable friendship", "attention"; and so on.

11541: <p>Consider the fact that our Nirvana depends on one crucial truth and that is law of Karma. Good karma bear good fruit and bad karma generates bad results. </p>

<p>Why didn't Buddha say that Fifth Noble Truth is law of Karma ?</p>
 471: <p>Kamma is already covered in the first noble truth that there is suffering.</p>

<p>From <a href="https://suttacentral.net/an5.57/en/bodhi" rel="nofollow noreferrer">AN 5.57</a>:</p>

<blockquote>
  <p>“This noble disciple reflects thus: ‘I am not the only one who is the
  owner of one’s kamma, the heir of one’s kamma; who has kamma as one’s
  origin, kamma as one’s relative, kamma as one’s resort; who will be
  the heir of whatever kamma, good or bad, that one does. All beings
  that come and go, that pass away and undergo rebirth, are owners of
  their kamma, heirs of their kamma; all have kamma as their origin,
  kamma as their relative, kamma as their resort; all will be heirs of
  whatever kamma, good or bad, that they do.’ As he often reflects on
  this theme, the path is generated. He pursues this path, develops it,
  and cultivates it. As he does so, the fetters are entirely abandoned
  and the underlying tendencies are uprooted.</p>
</blockquote>

<p>The noble disciple reflects that he is not the only one. He reflects that all beings are subject to kamma, which results in their birth, death and rebirth based on their kamma. This is one aspect of suffering. Reflecting thus, he chooses to follow the path laid out in the fourth noble truth, in order to become free from suffering.</p>

<p>But at the same time, we should not adopt the Jain fallacy that everything that happens is due to kamma, as seen in the <a href="https://www.accesstoinsight.org/tipitaka/sn/sn36/sn36.021.than.html" rel="nofollow noreferrer">Sivaka Sutta</a>:</p>

<blockquote>
  <p>"There are cases where some feelings arise based on phlegm... based on
  internal winds... based on a combination of bodily humors... from the
  change of the seasons... from uneven care of the body... from harsh
  treatment... from the result of kamma. You yourself should know how
  some feelings arise from the result of kamma. Even the world is agreed
  on how some feelings arise from the result of kamma. So any brahmans &amp;
  contemplatives who are of the doctrine &amp; view that whatever an
  individual feels — pleasure, pain, neither pleasure-nor-pain — is
  entirely caused by what was done before — slip past what they
  themselves know, slip past what is agreed on by the world. Therefore I
  say that those brahmans &amp; contemplatives are wrong."</p>
</blockquote>
 13806: <p>Good karma gives you good destination, and bad gives you bad destination. But it doesn't matter good or bad. All birth is suffering: because where ever born then there is aging and dying. So karma is in the second noble truth:</p>

<blockquote>
  <p><em>yāyaṃ taṇhā ponobbhavikā</em></p>
</blockquote>

<p><em>ponobbhavikā</em> ("leading to rebirth") is new karma. </p>
 43: <p>MN 19 <a href="https://www.accesstoinsight.org/tipitaka/mn/mn.019.than.html" rel="nofollow noreferrer">Dvedhavitakka Sutta</a>:</p>

<blockquote>
  <p>Before my self-awakening, back when I was still just an unawakened Bodhisatta, the thought occurred to me: What if I divide all my thoughts into two groups?' So I made thinking imbued with sensuality, negativity, harmfulness etc. one group, and thinking imbued with renunciation, non-negativity, &amp; harmlessness etc. - another group.</p>
</blockquote>

<p>So yes, if you understand karma as dividing all your action (physical, verbal, and mental) into wholesome and unwholesome, and then minimizing one and maximizing the other - then yes, this is a key teaching. And in a way, it's a complete teaching - one could follow it without asking further questions, and by figuring out all implications arrive at Buddha's result.</p>

<p>However, an explanation like "To avoid all evil, to cultivate good, and to cleanse one's mind — this is the teaching of the Buddhas." (Dhammapada) lacks context: WHY? Why do good? What's wrong with evil? What will I get if I follow this teaching?</p>

<p>So Four Noble Truths structure the teaching like a doctor would structure a consultation with the patient. First it discusses the symptoms, then the diagnosis (the root cause of the symptoms), then it states that removing the root cause will fix the problem, then it offers a methodical plan of healing.</p>

<p>The fourth truth, the Eighfold Path, is 100% based on the "two kinds of thoughts" and the idea of good vs. bad karma. Every item in Eightfold Path starts with the word Samma, which means "correctly done, good quality". So every item explains right and wrong, not just applied to Physical, Verbal, and Mental action - but also applied to View, Livelihood, Practice, Mindfulness, Meditation and so on.</p>

<p>Finally, as Buddha explained in MN 101, it is wrong view to assume that liberation happens as a result of good karma. Instead, according to Buddha's teaching in MN 101, liberation from dukkha is achieved by <strong>"abandoning of unskillful mental qualities and the attainment of skillful mental qualities in the here-&amp;-now"</strong>: </p>

<blockquote>
  <p>When I exert a [physical, verbal, or mental] fabrication against this cause of stress, then from the fabrication of exertion there is dispassion. When I look on with equanimity at that cause of stress, then from the development of equanimity there is dispassion.' So he exerts a fabrication against the cause of stress where there comes dispassion from the fabrication of exertion, and develops equanimity with regard to the cause of stress where there comes dispassion from the development of equanimity. Thus the stress coming from the cause of stress for which there is dispassion through the fabrication of exertion is exhausted &amp; the stress resulting from the cause of stress for which there is dispassion through the development of equanimity is exhausted ...  </p>
  
  <p>On seeing a form with the eye, he does not grasp at any theme or details by which — if he were to dwell without restraint over the faculty of the eye — evil, unskillful qualities such as greed or distress might assail him. On hearing a sound with the ear... On smelling an odor with the nose... On tasting a flavor with the tongue... On touching a tactile sensation with the body... On cognizing an idea with the intellect, he does not grasp at any theme or details by which — if he were to dwell without restraint over the faculty of the intellect — evil, unskillful qualities such as greed or distress might assail him. Endowed with this noble restraint over the sense faculties, he is inwardly sensitive to the pleasure of being blameless. ...  </p>
  
  <p>Abandoning covetousness with regard to the world, he dwells with an awareness devoid of covetousness. He cleanses his mind of covetousness. Abandoning ill will and anger, he dwells with an awareness devoid of ill will, sympathetic with the welfare of all living beings. He cleanses his mind of ill will and anger. Abandoning sloth and drowsiness, he dwells with an awareness devoid of sloth and drowsiness, mindful, alert, percipient of light. He cleanses his mind of sloth and drowsiness. Abandoning restlessness and anxiety, he dwells undisturbed, his mind inwardly stilled. He cleanses his mind of restlessness and anxiety. Abandoning uncertainty, he dwells having crossed over uncertainty, with no perplexity with regard to skillful mental qualities. He cleanses his mind of uncertainty. ...  </p>
  
  <p>Having abandoned these five hindrances — imperfections of awareness that weaken discernment — then, quite withdrawn from sensual pleasures, withdrawn from unskillful mental qualities, he enters and remains in the first jhana ... second jhana ... third jhana ... fourth jhana ...  </p>
  
  <p>With his mind thus concentrated, purified, and bright, unblemished, free from defects, pliant, malleable, steady, and attained to imperturbability, the monk directs and inclines it to the knowledge of the ending of the mental fermentations. He discerns, as it has come to be, that ... These are mental fermentations... This is the origination of fermentations... This is the cessation of fermentations... This is the way leading to the cessation of fermentations.' His heart, thus knowing, thus seeing, is released from the fermentation of sensuality, the fermentation of becoming, the fermentation of ignorance. With release, there is the knowledge, 'Released.' He discerns that 'Birth is ended, the holy life fulfilled, the task done. There is nothing further for this world. ...  </p>
  
  <p>Such is the teaching of the Tathagata.</p>
</blockquote>
 111: <p>Nibbana does not depend on anything. That's why it is called 'uncaused'.
Karma is covered in <a href="https://sites.google.com/site/rahulawhatthebuddha/the-second-noble-truth" rel="nofollow noreferrer">the Second Noble Truth</a>. Craving for sensual pleasures, craving for becoming and craving for non-becoming are all Karma.</p>
 11237: <p>The whole structure of the four truths can be seen as a teaching about karma (cause and effect); that's why the four truths are sometimes called the "two sets of cause and effect"; see, for example:</p>

<p>The third paragraph in <a href="https://www.accesstoinsight.org/lib/study/truths.html" rel="nofollow noreferrer"><em>The Four Noble Truths</em> by Theravadin teacher Thanissaro Bhikkhu</a>:</p>

<blockquote>
  <p>To counter this problem, the four noble truths drop ideas of me/not
  me, and being/not being, and replace them with two sets of variables:
  cause and effect, skillful and unskillful. In other words, there is
  the truth of stress and suffering (unskillful effect), the truth of
  the origination of stress (unskillful cause), the truth of the
  cessation of stress (skillful effect), and the truth of the path to
  the cessation of stress (skillful cause).</p>
</blockquote>

<p>The section titled "The Two Sets of Cause and Effect" in <a href="https://www.wisdompubs.org/book/four-noble-truths/selections" rel="nofollow noreferrer"><em>The Four Noble Truths</em> by Tibetan teacher Geshe Tashi Tsering</a>:</p>

<blockquote>
  <p>One of the key beliefs in Buddhism is that nothing comes into being
  without a cause, and this law of cause and effect is a fundamental
  point for understanding the four noble truths. The following table
  shows two sets of cause and effect: suffering as the result of the
  origin of suffering and cessation as the result of the path.</p>

<pre><code>               Result            Cause

Problem        1. Suffering      2. Origin

Solution       3. Cessation      4. Path
</code></pre>
  
  <p>Although these two sets of cause and effect are very specific, they
  follow the rule of all causes and effects, and in particular the type
  of cause and effect called karma in Buddhism.</p>
</blockquote>
 254: <p>Other answers say that kamma is already covered by, or implicit in, the first and/or second noble truths.</p>

<p>I think it's related to the <em>fourth</em> noble truth -- i.e. you say that kamma is important to attaining nirvana (and I think that "the path towards attaining nirvana" is what the fourth noble truth is about).</p>

<p>Or, perhaps, you can see "bad karma" and "good karma" in the second and third noble truths respectively.</p>

<hr>

<p>I find the logical structure of the <em>four</em> noble truths satisfying or persuasive.</p>

<p>The first three have the structure of a <a href="https://en.wikipedia.org/wiki/Syllogism#Basic_structure" rel="nofollow noreferrer">syllogism</a>:</p>

<ol>
<li>premise: dukkha exists, and is this and that</li>
<li>premise: dukkha is associated with craving</li>
<li>theorem: dukkha ceases when craving ceases</li>
</ol>

<p>So that's the doctrine "in a nutshell".</p>

<p>The fourth is, then, an introduction to the rest of Buddhism:</p>

<ol start="4">
<li>to begin to implement that in practice, see the noble eightfold way</li>
</ol>

<hr>

<p>I think that (as well as karma) there are other important aspects of dharma -- e.g. anatta, dependent origination, the sangha, the <a href="https://en.wikipedia.org/wiki/V%C4%ABrya#Pali_literature" rel="nofollow noreferrer">qualities conducive to Enlightenment</a>, and so on.</p>

<p>So I don't see why you see the need to "privilege" karma (as more important than other doctrines) -- but it's helpful that there is a doctrine with only 4 (memorable) truths, rather than (less memorable) 20. The "4 truths" served as (and still serve as) an introduction to the dharma.</p>

<hr>

<p>Or perhaps you're saying that the doctrine of karma should be privileged because it's the single most important doctrine to remember. If so, maybe you're not wrong; but there are also some other dharmas that people have been inclined to say are the most important -- for example, "right view"; or maybe "harmlessness", "generosity", "skilful virtue", "admirable friendship", "attention"; and so on.</p>
