I would guess your network has broken IPv6, which causes the problems (it's not a big
issue without Avast - except performance - because the browser normaly does a reconnect
to the servers IPv4 address which is not possible with Avast webshield enabled) If you disable
IPv6 support in the Avast webshield preferences, things will most probably get fixed.
If not, some more info is required, at least the relevant parts of the system log file.