Avast! doesn't block Gmail at all. I'm using it every single day, and many of other forum members do too.
Why don't you install Firefox, and then go to your online banking ? avast! doesn't have script blocker for Firefox, so you'll see where is your problem...
Also, Script bloker is not the one that blocks that stuff... it's a web shield. If you are using ZoneAlarm Pro as your firewall, web shield could give you some troubles accessing some sites, because guys from ZoneLabs still didn't do anything regarding incompatibility issues with avast! web shield. You still can go through proxies or completely disable avast! web shield. Search these forums for advice on how to do that...
Cheers !