The point is not to disable it when installing the application, but to disable it before running the application if there is some connection issue. This is a common issue especially with third-party anti-virus applications like Avast. Completely disabling the anti-virus or firewall application means that eM Client will have unrestricted access to the Internet and you can test it. Often users will only partly disable Avast, and that will not work. But maybe @Charlie6 did not include everything. Maybe it is your firewall.
Obviously when you started in Safe mode and it worked, that means there is something that is running in normal mode that is blocking eM Client from accessing the Internet as @cyberzork says. This is troubleshooting 101, but I think that the problem is you have elevated yourself to such a height that you cannot see the ground anymore.