You will need to buy a retail copy of Tiger (10.4) on a disc, if you want to upgrade beyond Panther (10.3.9). But before you do anything, you should state the specific model of PowerBook you have, in order to determine what version of Mac OS X is the last verion you can run. Some later PowerBook with a G4 can run Leopard (10.5).
These lists show all the PowerBook G3 and G4 models, if that helps to identify your PowerBook
With 10.3.9, these versions of various web browsers should work. They are mostly older versions, but not nearly as old as Safari for 10.3.9.
Camino 1.6 - http://caminobrowser.org/releases/1.6.11/
iCab (latest version) - http://www.icab.de/dl.php
I am having this exact issue and stumbled across your solution... However... when I tried to download any of these earlier versions Safari crashes... is there any workaround for this problem? I have been spinning my wheels for days and it seems like the old iMac G5 is going to be more of a dust collector than anything else... I can tell you that this particular iMac is running OS X 10.3.9 with Safari 1.3.2. I have even gone so far as completely erasing the hard drive and reinstalling OS X
You could download one of the other browsers (or all of them) on another computer, copy the download(s) to a USB flash drive (or burn them to a CD), then copy to and install on iMac.
If Mail is working on the iMac, you could send (to yourself) one of the browser downloads (the smallest one) from another computer, as an email attachment. Once that one browser is installed, you can hopefully download and install the others.
Although Safari from 10.3.9 is old, it should be able to download a file without crashing. There may be some other problem with your iMac, or your current installed system.