Safe mode in Windows installs only the OS and the drivers for mouse, keyboard, and standard VGA mode. It does not load the commercial graphics driver, printer/scanner drivers, etc. It also does not load most executables from apps and other such software.
You mentioned you had installed Catalyst Control Center (related to your video drivers), so it's possible something in that install has led to a conflict with Opera and Chrome that other browser engine types don't encounter. CCC enables all manner of custom display modes for the ATI Radeon graphics chips/cards, and that may be where a conflict originates.