How exactly do the browsers fail to load the websites? Do they crash? Is there an error message? Do they even start to load? Do you have any other security software installed? I'm assuming a full system scan came up clean?
Since it happens with both browsers and works while in Safe Mode, it's caused by a third-party. Try checking Task Manager (ctrl + alt + delete) and find out what processes are running. It should give you some indication of what's causing the problem, and the above questions should help explain how it's doing it.
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Win XP, Dell XPS machine. IE7 (beta), IE6 and Mozilla fail to open most web pages after a normal startup. (No IE6 and 7 were not installed at the same time). I have tried disabling, my Anti-Hacker, Anti-Virus (Kaspersky) without positive results. When i start the computer in safe-mode with networking the web pages open. With a normal startup, it seems to only open secure websites, ie banking etc. Any hint as to why this is occuring?