"So I filled out a removal-excess-form to remove the excess contribution of $2,400 for 2022 from my traditional IRA in one or two days after the recharacterization,, and received $2,400."
It seems that the financial institution miscalculated the attributable gain. The gain or loss should have included the gain while the $6,000 contribution was in the Roth IRA. Given the short time between the recharacterization and the return of contribution, the return of the $2,400 of contribution should have included about $960 of gains. (The start of the computation period for the gains should be the date of the original Roth IRA contribution because the recharacterization treats the original Roth IRA contribution as having been a traditional IRA contribution on that date.) It will be difficult to provide an appropriate explanation of the return of contribution if the gain was not calculated correctly. I would try to confirm with the financial institution the details (particularly boxes 2a, the marking of either of the boxes in box 2b and the code in box 7) of the 2023 Form 1099-R that you'll be receiving in early 2024.
If one treats the distribution of $2,400 as satisfying a return of $2,400 of contribution (which I disagree with because I believe that the attributable gain was not correctly calculated), it means that you have only contributed $3,600 to a traditional IRA for 2022. TurboTax does not have a good way to deal the complexity of a return of contribution from a recharacterized contribution, you might need to just enter the resulting $3,600 traditional IRA contribution instead of entering the original Roth IRA contribution, indicating that you switch the contribution to Roth , then indicating that you received a return of a portion of the contribution.
