My employer paid $5250 out of my total $6422 tuition expenses. How can I only claim the amount that I paid?

Box 2 of form 1098-T shows the amount billed by the school and this is not the amount used to claim education credits. This is why TurboTax no longer asks for the amount in box 2.

The amount used for claiming education are qualified education expenses you actually paid to the school in 2021 which should be shown in box 1 of form 1098-T.

If your box 1 does not show the correct amount paid, when filling out your form 1098-T in TurboTax, just enter the amount in box 1 of your form 1098-T on the entry screen.

Under box 1, on the 1098-T screen is a radio button: This is not what (student name) paid to this school. Click on that and you will be given a chance to enter what was actually paid in 2021 by excluding the portion paid by your employer.

MinhT1's   workaround will accomplish what you need and is probably simpler that the way TurboTax (TT) is designed to handle.

Alternatively, it can (and should) be done this way:

Enter the 1098-T.  Note whether the school has included the $5250 reimbursement in box 5 (they shouldn't but some do).  If so, that should cover it.  If not, a few screens later you will be asked if you got any employer reimbursement.  Enter only the tax free reimbursement.  That entry will allow TT to calculate the credit on the $1172 difference.