Had the same problem. Just spoke to Apple Support on the phone to fix it.
Try creating a temporary new user account in System Preferences. Log in with that user, start Safari and try iCloud.com. If this looks good (for me it did) then your problem is in your user account.
- I tried Empty Cache in de Developer Menu. Didn't fix it.
- I tried resetting Safari. Didn't fix it.
- I tried removing the Library Safari folder and Safari Preference files. Didn't fix it.
What did work:
1. Restart your Mac and boot in Safe Mode (hold shift).
2. Login with your regular user account.
3. Start Safari and go to iCloud.com. I got a flickering screen and iCloud was even more unusable. But...
4. Restart your Mac in normal mode.
5. Open Safari and login to iCloud.com
Booting in Safe Mode seems to reset Safari in a different way. I'm not sure step 3 is needed, but just in case.
Good luck!