If your Mac only works normally in Safe Mode, a third party app is causing the crash. There are a few things you could try here.
First, create a new user account and see if the problematic app works fine there. If it’s a third-party app, I suggest completely uninstalling it. Delete everything - including system files. You need to go to the ~/Library folder and follow through to Application Support - any folders with the app’s name must be deleted. Caches, too. Don’t forget to empty Trash.
Second, attempt repairing your drive in macOS Recovery Mode. Restart your Mac and hold down Command + R as it boots. From the Utilities window that appears select “Disk Utility”. In the Disk Utility window, click on Macintosh HD in the sidebar on the left. Then click on First Aid. This should automatically detect and repair any problems with your drive. Keep running First Aid until no problems are detected.
If you keep experiencing the problems, definitely reinstall macOS. You mentioned that it won’t let you do that - have you tried wiping your hard drive in Recovery mode Disk Utility first? You can try starting up the Mac from an external drive as well - see if that makes any difference.