If your school has determined that you qualify for an educational credit, then they should issue you a form 1098-T. Lacking a 1098-T, from the school, you will still need the school's federal Employer identification number (EIN).
A "letter" may be of some help if you get audited, but probably not.
That said, you ask "then why is TurboTax (TT) saying otherwise?" It's because you're not giving the right answers to the TT interview questions. For one, when TT asks if you have a 1098-T and you answer no, you must answer yes to the do you qualify for an exception question. That gets you to the rest of the interview, where you have more questions that must be answered correctly.
To be eligible for the tuition credits, the course must be taken at "an eligible institution". The school should be able to tell you if it is an eligible educational institution. In general, an eligible educational institution is an accredited college, university, vocational school, or other postsecondary educational institution, including accredited, public, nonprofit, and proprietary (privately-owned, profit-making) postsecondary institutions. Additionally, in order to be an eligible educational institution, the school must be eligible to participate in a student aid program administered by the Department of Education. If they issue a 1098-T they are probably an eligible institution.
Enter your school at the link below, to see if it's on the dept. of education list.
https://ope.ed.gov/dapip/#/home
Rule of thumb: if you have to ask, your school is not an eligible institution.
Schools that only offer online courses usually do not qualify. Online classes taken at schools that also have classroom instruction usually do count.
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 a 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
