TurboTax should automatically carry over the amounts to 2023 but it would be a $0 2022 conversion contribution taxable and a $5,200 2022 conversion not taxable (since the conversion amount was $5,200 and you should have a basis of $800 left on line 14 of Form 8606 to carry over to 2023).
Yes, if you only made a $6,000 direct nondeductible traditional IRA contribution for 2022 and had gained $500 before the conversion and it would be a $500 2022 conversion contribution taxable and a $6,000 2022 conversion not taxable.
You will be able to find this information on your IRA information worksheet Part V (balance carryover to 2023).
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