I don't think the conflict is with Catalina. Amazon shows over 1,000 ratings for TT 2020 with a mean rating of 4.4. I have been using TT 2020 on Catalina 10.15.7. If there was a fundamental conflict that was with the MacOS, it would not work for all users of 10.15.7, which does not seem to be the case. That said, what is happening to you is real, it is more likely some conflicts with something you could have installed earlier. If you download and run the free diagnostic program Etrecheck (written by a trusted Mac expert who provides advice in these Discussions) and post the results here via the small paper icon in the lower right of the "Reply" box for these Discussions, the Etrecheck developer and others may be able to identify the item causing the conflict.
One thing to try -- boot into Safe Mode, holding SHIFT key while restarting your computer. This may take a long time, 10 minutes, maybe even more. One thing that Safe Mode does do besides repair the file system, is to clear out some caches. In some instances, a leftover cache file can interfere with the proper operation of TurboTax. [I don't know if TurboTax will run in Safe Mode, but it is something to try while you are booted there, otherwise wait for Safe Mode to boot, then restart normally and test again.].
Another thing to try: log out and log in as another user (you can create one if you don't haver one already created) and try running TurboTax as that other user. If it works for the other user, then it would be something specific to your original user that conflicts. If the other user has the same problem, that would unfortunately mean that the problem is system wide, not as easy to address.
Another possibility: if you have any anti-virus, non-Apple security software, or "app cleaner" or "optimizer" type software running on your Mac, try uninstalling them, they may be conflicting as they often do.
Lastly, here is something I used to do when my TT 2019 sometimes did not run properly on my 2011 MacBook Air: empty the trash; then shutdown; wait a few seconds, then restart, and don't open anything except TurboTax. This always worked for me on that older computer.