The newest Radeon graphic drivers are not working correctly with SimCity. The "fix" that seems to work is to install an older driver to use with SimCity.
The last driver reported to fix it was non-whql-win7-64bit-radeon-software-crimson-relive-17.6.2-june13, that's obviously really old. You could try installing a stable one from last year to see if that fixes it. If you find a newer one that works well, report back so others can know what works.
(I know that installing an older driver is not an optimal solution. It's just that SimCity is an older game and EA is no longer issuing updates to it, so it will not be something they will fix. You could try seeing if AMD will do something to support SimCity and issue a fix for their newest cards and drivers.)