Hello and welcome to the support community.
Which exact iBook G4 model do you own?
If you go to the apple logo top left and then to 'about this mac' let us know what it says.
Is it also the 12" or 14" iBook model?
I know what you mean with the unexpected quitting of Safari and FireFox under Mac OS X 10.3. My iBook G4 does the same. I resorted to installing tenforfox and camino
Sadly your not resolve the flash problem with any browser you use. Flash for Mac PPC (G4 iBook's) is very much out of date and can't support the current flash websites like YouTube. This would be the same for updating the OS.
You can upgrade your iBook depending on the specifications. If you let us know which version of the iBook you have, we could advise you to which would be the best version to upgrade to.