I'm using Thunderbird 38.1.0.5666 on Windows 7 Professional (32-bit), and I have not had any problems with Mozilla Thunderbird crashing.
However, I chose not to install all of the extras offered when Thunderbird was going through setup. Unfortunately, I do not remember exactly what those items were... chat... calendar? I also have Mozilla Maintenance Service disabled and update the program manually through the About->Check for Updates button.
I also did the same thing with Avast! and only have the minimum installed (not all of the extra features offered during setup).
If Thunderbird only works in Safe Mode, you have to find out all the things that aren't being loaded... then go one by one to see what additional features are causing the program to crash.
The problem might not completely be with Avast! but with Thunderbird. You had mentioned uninstalling Avast! and getting Thunderbird to work. Try disabling Avast! to see if that makes any difference. You can also try disabling each of the Avast! components (go to the Active Components section in the settings) to see if that makes any difference.