If my 1098-T doesn't report all of the out of pocket tuition paid to the school, how do I include it?

To verify, you are saying you received more scholarships/grants than listed in box 5? But I assume you meant that you paid more than listed in box 1. If this is correct then please follow these steps to enter the total tuition paid:

1. Login to your TurboTax Account 
2. Click on "Search" on the top right and type “1098T”
3. Click on “Jump to 1098T”
4. Click the link "This is not what I paid to this school" under box 1 on the 1098-T screen and enter the proper amount in the box on the right that will appear. 

If you received other scholarships/grants not listed in box 5 then you can enter them during the interview on the "Did You Receive a Scholarship or Grant in 2022?" screen

The 1098-T is only an informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or possibly your student has taxable scholarship income. 

If you claim the tuition credit, you do need to report that you got one or that you qualify for an exception (the TurboTax interview will handle this)

You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T. In the 1098-T screen, click on the link "What if this is not what I paid the school" underneath box 1. You will then be able to enter the actual amounts paid. 

Or if you find it easier, just change the numbers in boxes 1& 5 to what your records show. The 1098-T that you enter in TT is not sent to the IRS.