How to update browser with Panther 10.3?

I have just upgraded my ibook to Panther 10.3. It is working fabulously. But there is one problem.. A lot of the websites I try to go on don't appear normally.. Even Facebook. Is this because of the browser? If so what can I do because I looked at the newest version of Safari and it doesn't look as though it is compatible with Panther and I don't want to install ANYTHING that is not compatible.. Can anyone help me? I ran a software update, but that doesn't seem to have done anything with this.