Let me explain what you are seeing:
"Employer contributions were taxed in 2022, and were subsequently withdrawn." TurboTax told you that you had excess HSA contributions, and you withdrew them per the instructions. The excess was added to your Other Income, and you withdrew the excess to avoid a 6% penalty on the excess carryover to next year.
"TurboTax 2023 asks for the amount reported on 2022 Form 8889, which was the amount included in income in 2022." No, it didn't; it asked for the amount on line 48 on your 2022 form 5329. And since you withdrew the excess in early 2023, there shouldn't be an amount on line 48, so no double taxation.
"As soon as that amount was added, 2023 tax increased." Sure it did, because you shouldn't have entered anything. Please go back and read that input screen carefully.
And if you indeed had an amount on line 48 on the 5329, come back and tell us, because this doesn't match what you told us up above.
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