There could be some issues with your internet. Try to change the browser which you are using or clean the cache. Sometimes it creates issues like not loading, slow internet, or similar things like that. I hope it helps you.
I checked your account and it seems like the messages section is working. I'd suggest that you please check and make sure that your internet connection is stable. You may also try to use a different device if it's available so that you can check further. You may also want to try clearing your Upwork cache and cookies. If the issue persists please let us know.