contributed 13500 to IRA during 2024. (6500 from 2023, 7000 from 2024). Then did a backdoor conversion. Now I'm struggling to show that those 6500 belong to 2023.

Roth contributions spread across 2 years are a 2-step process.  For the 2023 tax return, you should have prepared a Form 8606 - check your tax return to verify you did.  If not, you can do so now.  There may be a $50 late fee assessed unless you have cause.  See: About Form 8606, Nondeductible IRAs to get a pdf form and fill it out, if needed. The 2024 8606 will then have the 2023 contribution reported on line 2 of the 8606.

Next, since the conversion for both years happened in 2024, it is correct that they were both reported to you on the 2024 1099-R.  In order to ensure no tax is computed on the conversion, since you did not deduct them in the first place, after you enter the 1099-R, continue through the interview until you are asked f you moved the money through a rollover or conversion. Answer that you converted (all of it) to a Roth IRA.