You are mixing up ideas: The web ≠ The internet.
Some Possible problems
There could be a problem with:
A web proxy at your ISP.
The DNS at your ISP.
To test DNS, in a shell (if on MS-Windows use CMD)...
You don't say which platform, but for Linux (i.e. Ubuntu) from this answer to How to easily get all HTTPS addresses that an application connects to externally?:
You can use lsof and watch to do this, ...