Can I take online courses to make up for poor past community college performance when applying to a four year college? I am wondering if there is a place where I can take online mathematics courses that do not require a transcript for admission. I did ok in high school and not so great at community college, but I found a great math tutor, and I'm willing to put in a lot more effort to get a degree. Before meeting this tutor, I was conditionally accepted to a four-year institution for mechanical engineering and was kicked out after the first semester. My plan is to meet with this tutor for about 20 hours a week until I feel comfortable with material up to calculus, then take a graded online course in calculus, so I have evidence of my understanding of calculus and my ability to excel in classes at a four-year university. I would repeat this process for multivariable calculus, differential equations, linear algebra, applied statistics, numerical analysis, abstract algebra, and real analysis.

I looked at Coursera, and they have calculus, differential equations, linear algebra, numerical analysis, and complex analysis. Do you think that taking these courses is the best chance that I have to re-enter my four-year institution for mathematics instead of engineering? I am concerned because the grade is based on just four quizzes, which means that it might not be recognized by undergraduate schools. If not, what would my best next step towards this goal?

Things are tough I need good grades to enroll at a university to show that I am qualified despite my previous bad performance, but I need to enroll in a university to take advanced classes to get those good grades.
My advice would be to go to the website of the university you are thinking about enrolling in, find out who the course coordinator is, and email them about your situation.  Explain that you are willing to put in the effort to get your knowledge up to an acceptable level.  You may find that they can offer you a bridging course, or suggest having sessions with a personal tutor like you are planning.

I found myself as tutoring a mature-age student who was very much in a situation similar to what you described.  The course coordinator emailed the maths academics to see if anyone was interested in some private one-on-one tutoring, and I took up the offer.  Ultimately, the student opted to drop out due to the feeling of being overwhelmed with the content (and they were struggling greatly to understand basic high-school level concepts), but the fact is the university attempted give the student every possible chance of getting in. There is nothing to lose by trying.