I attended 4 universities in 2020. At one I was a full-time student and took one class at the others. How do I file multiple 1098-T forms?

Yes. TurboTax can handle this. After completing the interview, you will be asked if the student went to another school. Answer yes. After completing the 2nd 1098-T, your will be asked (again)  if the student went to another school. Answer yes.  Payments to  all schools counts toward the credit, as long as the student  was an eligible student at his main school.

" I paid out of pocket for the courses I took at the other three colleges".  Money paid  from student loans also courts as paid by you.  There's even a loop hole available to count scholarship money for a tuition credit.

The student reports all his scholarship, up to the amount needed to claim the American opportunity credit, as income on his return. That way, the parents  (or himself*, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship.  You cannot do this if the school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.

Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 1. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.

*A full time unmarried student, under age 24, is only eligible for the refundable portion of the American Opportunity Credit if he supports himself by working. You cannot be supporting yourself on parental support, 529 plans or student loans & grants. You usually must have actually paid tuition, not had it paid by scholarships & grants.  It is usually best if the parent claims that credit. 

You cannot claim a credit if you are, or can be, claimed as a dependent by someone else.