Hello there, and sorry for not responding earlier.
I finally found a solution, it was just one click away, even though it cost me some days to find it. I use the theme Sonaar, and I had to switch off Ajax in the main settings of the theme. Now everything works fine, even another problem I had with the site was solved. The functionality of the site is fully given even without Ajax, and the TranslatePress Plugin works perfectly fine now. Problem Resolved 🙂
Thanks again and greetings!