I'm having the same problem on a site that has 2 domains that share cookies through an <iframe>. Up until now disabling "Prevent cross-site tracking" has always allowed the site to work. Now with Safari 13.0.4, it's like the option does not disable even when you uncheck the preference. This is failing on High Sierra and Mojave after Safari was updated, but oddly enough it seems to work on Catalina with the limited testing I did.
Toggling the option on and off as suggested in the Reddit thread did not work for my site.
Chrome and FireFox continue to work fine when the appropriate 3rd party cookie sharing options are enabled.
This version of Safari is seriously broken and there is no way to go back. You can't restore a previous version of Safari from Time Machine without restoring the entire OS. These are serious design failures.
Edit: I should add that I was updating from Safari 12, so I cannot verify if 13.0.3 would have worked or not.
Mac Pro (Late 2013)
High Sierra 10.13.6
Safari 13.0.4