How can I claim the American Opportunity Credit? TurboTax says I'm eligible for $1000 but it doesn't show on my return amount.

Where are you looking that it doesn't show on your return amount?

If TurboTax is saying you are eligible for $1,000 you can look at your form 1040 line 29.  You should see the $1,000 there.  You can also see it on line 3 of your Schedule 3.

1. From the left rail menu in TurboTax Online, select Tax Tools (You may have to scroll down on the left rail menu.)
2. Select Tax Tools 
   
3. On the drop-down select Tools 
4. On the pop-up menu titled “Tools Center”, select View Tax Summary 
5. On the left sidebar, select Preview my 1040 

The error may have been  earlier in the TurboTax (TT) program when TT may have  indicated you might be eligible for $1000. 

There's a new urban myth among college students that says they can get a $1000 from the government just for filing a tax form. For most of them, they simply aren't eligible. A full time unmarried student, under age 24, even if you don't qualify as a dependent, 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. It is usually best if the parent claims that credit. 

You cannot claim the (up to) $1000 refundable credit if you are, or can be, claimed as a dependent by someone else.

Reference: Line 7 instructions for form 8863.

https://www.irs.gov/instructions/i8863#en_US_2024_publink53002gd0e674

If you are eligible, TT by passes Schedule 3 and takes the amount from line 8 of form 8863 and puts it on line 9 of form 1040.