Upgrade from Safari 4.1.3
Safari 4.1.3 is the last version of Safari for Mac OS X Tiger, so you're out of luck on that end. Furthermore, Chrome and Firefox (in their current versions) do not support Tiger either.

You can try looking for Mac OS X Leopard, since Safari 5.1.10 (the last version of Safari for Leopard) is available for download, but that's as far as your computer will possibly be able to go, since the latest version requires, at minimum, OS X Mountain Lion, which requires an Intel processor.

This Apple Discussions page suggests you could also try TenFourFox.

If you don't like any of these options, you'll have to purchase a new Mac.