If I transferred my Roth IRA to a new provider in 2020, and also contributed the allowed $6,000 to it, how do I reflect that in TurboTax?

If done with a trustee-to-trustee transfer then there is nothing to report.   If you received a distribution and rolled it over yourself then you will receive a 1099-R that you say was rolled over when you enter it.

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.

These are two entirely separate transactions.  The IRA contribution is entered under Deductions & Credits -> Retirement and Investments -> Traditional and Roth IRA Contributions.  Nothing regarding the movement of the Roth IRA from the old provider is entered there.

If the Roth IRA was properly moved by trustee-to-trustee transfer where the funds were never paid to you, there is nothing to report regarding this transfer.  However, if the original Roth IRA was distributed to you and you rolled it over to the new Roth IRA, generally within 60 days, resulting in the old custodian issuing a Form 1099-R reporting a distribution from the original Roth IRA, enter code J, T or Q Form 1099-R under Wages & Income (or Personal Income) -> Retirement Plans and Social Security -> IRA, 401(k), Pension Plan Withdrawals (1099-R), indicate that you moved the money to another retirement account, then indicate the amount rolled over.