Fedora 5 is fairly old as Fedora goes is a 2.6 kernel version so isn't ancient. If you have the cycles you might want to consider upgrading to Fedora 8 (they've dropped the "Core" from name as of Fedora 7).
Notes I wrote about upgrading Firefox (or Thunderbird):
To install Firefox or Thunderbird on Linux:
Download the tar.gz package from www.mozilla.org
Look in /usr/lib to see if an earlier package exists - if it doesn't have version number suffix you may need to save it.
Unbundle the package to /usr/lib:
tar xzvf /root/firefox.184.108.40.206.tar.gz
This should create a directory with the install usually without the version.
So you'll know later you should renamed the directory with the version:
mv /usr/lib/firefox /usr/lib/firefox.220.127.116.11
Make a link (check for existing executable first and save it if desired):
mv /usr/bin/firefox /usr/bin/firefox.old
ln -s /usr/lib/firefox.18.104.22.168/firefox /usr/bin/firefox
At that point you should just be able to start the application by typing the command at the prompt:
Also the GUI "world" icon is for the browser and should already point to /usr/bin/firefox. If so you should be able to start firefox after the above procedure by clicking on it as you did before. You can go to Help - About Firefox after you start it to verify you have the latest version.