I've installed KDE on my Arch system while being logged in as root.
When I log in as root - everything seems okay, but when I log in as a simple user - the desktop is broken and after a 20-50 seconds it turns black with only cursor available (after I've played a little bit under root user it stopped turning black for a simple user, but desktop is still broken).
Here is how it looks for a simple user



Also, when I click the power button (a physical button) a nice screen appears that allows me to shutdown, reboot, log out etc. and it looks okay, so it's only the desktop issue.

How do I fix it for a simple user? What should I be looking for?
Thanks.

You might check for permissions issues in the user's home directory. In a terminal, you can run "ls -al" to see the permissions of all of the files in that directory (add -R to make it recursive, pipe it into grep to filter the results: "ls -Ral | grep -v <username>:<username>"). If anything shows up as not being owned by your user then you can use chown (as root) to fix the permissions.

Likely culprits would be stuff in ~/.config or ~/.local where most of the configs and user-local settings are stored.

You can also create a new user and see if you have the same problems under that user.