Is not being paid in an internship common in academia? I'm a to-be third year Phyiscs and Mathematics student, and in the beginning of this summer, I was invited to do internship at the lab of one of the professors in the Physics department by that professor herself, and I accepted directly.

During this last 1 month, i.e since the beginning of my internship, I have basically invalidated the work of 2 master student, 1 Phd student and a undergrad student's joint work as a whole (the methods that they were using to analyse the data methodically insufficient and wrong, and I have shown this), and solved the problems that they couldn't solve, and got consistent results.Further, I have made suggestions for how we can improve our result about the experiment that the lab is conducting.

Now, this is my first internship that I have ever done, and I did (and I think I still do) not have any idea about internship in general, and I did not asked for any allowance for the work that I was going to do, and just interested with the job, and I have learned a lot of (literally a lot) things in this 1 month, and enjoyed with the work I was doing.

The thing is, I cannot take any course credit from an internship (this is a thing in my University), and I did not get paid; it is OK, I have learned a lot of things, but I'm asking just for the future internships, is this kind of situation common in academia in general ? Because, after some time the situation transformed from helping researchers in their research to doing the very work of those researchers that they cannot do properly while they still got paid.

Edit:

I'm in Turkey.
Sadly, yes. It is all too common, though not universal. Not only for internships in labs, but also in businesses. The US Congress, even, doesn't pay its interns, though there is some move to do so in future. 

The "justification" of this is that the students are "learning", which can be true or false. But normally they do work for which pay would otherwise be required. 

In my opinion, this is just wrong. If people are contributing, they should be paid. If they are not paid then many people who need to support themselves - somehow - will be closed out of these opportunities. I would have a bit less objection to it but for this fact. In economically diverse countries such as the US (and others) it is especially important. 

The solution to it is political, however, whether at the university level or the governmental level. 

As you say, of course, an internship teaches you things that are hard to learn otherwise. They are valuable for themselves. Your CV can reflect internships you have done. 

Well, some internships teach you things. There are also examples in which "interns" become bottle washers, not learning much of anything. This is pure exploitation. 

But labs, agencies, and businesses need to include sufficient funding so that students don't get exploited.