Are you sure your Mac is compatible with Catalina? You didn’t mention your model. Early Mac models from before 2012 don’t meet the graphics cards requirements for the latest version of the OS. Here’s a full list of Macs compatible with Catalina. If that’s not the issue, you should first try restarting your Mac. If that doesn’t help either, you might have a problem with your internal drive. You might need to have a clean reinstallation of the OS. You’ll need to start up your Mac in recovery mode. Here’s what to do: 1. Restart your Mac and immediately hold down Command + R as it’s booting until the startup screen appears; 2. When you see the Utilities window, first thing you should do is to wipe and reformat your disk before you can reinstall the OS. Bear in mind all of the data on your Mac will be cleared, so make sure you’ve backed it up. Go to Disk Utility; 3. Click on View → Show all devices in the top left corner. Delete the Macintosh HD-Data volume. Click on Macintosh HD, unmount it and click Erase. Choose APFS for format, click Erase again. Enter your Apple ID; Quit Disk Utility and go back to the Utilities screen; 4. Choose “Reinstall macOS” - follow the instructions and reinstall Catalina.
Hope this helps.