Your financial institution would have reported the contributions on a Form 5498 sent to you and to the IRS each year that you made a contribution. These would be the only records beyond your account statements. I'm not sure how far back you can go in obtaining Wage & Income transcripts from the IRS which would include these.
If you haven't withdrawn any of the money, TurboTax doesn't need to know this information. But you might need it in the future if you withdraw any of your Roth IRA money before age 59½.