Obviously something's getting started in normal mode that's not in Safe Mode, so here's what I'd do. Hold down the Windows key and press R. Then type in MSCONFIG and hit enter. Click the Startup tab. There you'll see a list of tasks that're being started automatically when you boot up in normal mode. You can uncheck anything there except your antivirus program, so do that then reboot. If that fixes the problem, you can start adding things back 2 or 3 at a time until you find the culprit.
If that doesn't resolve the problem, you can check for corrupted system files. Open an administrator command prompt and run SFC if the above doesn't help. Click START, then type CMD in the search box, right-click CMD.EXE and click Run as administrator. Then from the command prompt type sfc /scannow.
As already stated it will not run .exe files in normal mode, but works perfectly fine in safe mode. I have used two programs, McAfee and Malwarebytes, but no viruses were found. I was suggested Kasperskys but I cannot install this in safe mode.
Also, anything that I do manage to successfully open in normal mode either freezes or stops responding for long periods of time.
Additional information if necessary:
Model: Satellite C655D
System type: 64-bit operating system
196 GB free out of 284