First of all, as we've found out in the comments, you actually use Premiere Rush and not Premiere Pro. There is a difference, Premiere Pro is much more complex but it also enables you to take control of many more aspects of the video making process.
The problem is most likely due to compression with a target bitrate that's too low. Rush is optimized for quick social media video sharing and this means they expect videos to be about a person talking about something filmed in front of a mostly static background, for example. They didn't have videos as yours in mind that require higher bitrates to look sharp and clean.
Normally I don't use Rush, but tried it to check what can be done in your case. Unfortunately, I don't see any option you can try. Setting the quality to high is already the best you can do. Using the 4k preset may help if the original clip has been recorded at 4k.
Another option you can try is to install ffmpeg and use it from the command line. A question that deals with adding the crop effect, you need, is this one: How can I crop a video with ffmpeg?
Handbrake is a GUI tool, that uses ffmpeg and another alternative to try.
Or you could try Premiere Pro (if it's not in your subcription plan you could still try it for a number of days for free). However, both Premiere and ffmpeg are more complex.
It might be good to record another 5 min clip to save time when you try to find out the highest bitrate you need to make it look pleasant, yet still produce videos that are suitably small for upload to the web.