Leave the file system shield on and try disabling the web shield as that is the shield which would have a direct interaction with web browsing.
The file system shield doesn't monitor/scan web traffic, so I don't know why there would be any slowdown.
Have (or did) you another Anti-Virus installed in this system, if so what was it and how did you get rid of it ?
Have you just installed avast ?
What other security applications do you have installed, firewall, anti-spyware, etc. ?