It can take a while. If you ever get to the log in and manage to log in then it generally starts okay each time though sometimes doesn’t.
Turning TV off and on again may help. Also, if you haven’t tried it, update the TV firmware. May or may not help.
However the Amazon app on Smart TV is a bit rubbish. Most other apps inc Netflix & Plex are fine, but Amazon seem to have crippled the Smart TV app. Actually the Red Bull app is garbage on Smart TV also.
Main problem I have though with the Amazon app is audio drop out on 5.1, which is a known issue for years and never fixed.
Their app on Fire TV stick/box is of course perfectly fine, but then they want you to buy it 😉