Yes, turbotax can handle this situation. You already seem to know the rules. Each year that you make a partial conversion, you will generate a form 8606 that documents the non-deductible basis in the traditional IRA at the beginning and end of the year (so the year end number from 2024 is the beginning number for 2025, and so on) and calculates the tax due on the conversion. You will want to keep copies of the form 8606 for as long as you have any basis in a traditional IRA, plus 6 years after. (This is an exception to the general rule that you can throw out tax papers after 3 or 6 years.)
You are correct that the broker will not know the taxable amount, that is what form 8606 is for. (The broker would generally not report the taxable portion in any case, because the pro rata rule aggregates all your traditional IRA balances into one calculation, and broker A won't know whether or not you have other accounts with broker B, C etc.)
Basically, Turbotax can handle this. You should always review your return for accuracy before you file, and you will want to pay close attention to form 8606.
