I contributed $5,760 into my Roth IRA from 2020 to 2023. I took an early withdrawal of $5,000 last year. Do I have to pay taxes and penalties on this amount?

You should not have to pay taxes on the withdrawal of Roth contributions only(not earning).

You should have received a 1099-R. ‌Here is how to report it in TurboTax Online:

1. Go to Wages & Income.
2. On the Your income and expenses screen, locate and select the Retirement Plans and Social Security section, then select Start or Revisit next to IRA, 401(k), Pension Plan Withdrawals (1099-R).
3. Answer Yes to Did you get a 1099-R in 2024?, then Continue.
4. Select how you want to enter your 1099-R (import or choose Change how I enter my form to either upload it or type it in yourself) and follow the instructions.
5. Answer the questions on the following screens, until you reach Enter Prior Year Roth IRA Contributions. Here, enter all the amounts you contributed prior to 2024.