It does if you always transfer the prior year to the current year and enter the Roth contribution in the IRA contributions section and answer yes to the question about tracking your contributions.
I have been doing that for about 15 years and the total is always carried forward.
(If you skip transferring a year the carry-forward will be lost.)
Enter IRA contributions here:
Federal Taxes,
Deductions & Credits,
I’ll choose what I work on (if that screen comes up),
Retirement & Investments,
Traditional & Roth IRA contribution.
OR Use the "Tools" menu (if online version under My Account) and then "Search Topics" for "ira contributions" which will take you to the same place.
A Roth IRA contribution does not actually go on a tax return, but you should enter it anyway to:
1) Tell you if your income qualifies you for a contribution and warn you if it does not.
2) Check if your income exceeds the limit to contribute to a Roth.
3) Track your contribution year-to-year if you use TurboTax every year.
4) Add the Retirement Savers Credit if you qualify.