How can I update my Mac OS X 10.5.8?

If your computer is PowerPC Mac (G3/G4) portable, you could try TenFourFox browser.

Or if your computer is early Intel-based Mac, with 10.5.8 or 10.6.8, you could adapt a

version of TenFourFox (see the TenFourFox-G3, instructions for use with Intel-mac).

I've late model PowerPC G4 macs (desktop and portable) that run 10.4.11 and 10.5.8;

plus one early MacBook1.1 coreduo 1.83GHz (2006) and these all use TenFourFox. In

fact the intel-based MacBook with coreduo and 2GB RAM (upgraded from stock) is

fast; helps the rotational HDD runs at 7200-RPM. Much quicker than PPC G4 iBook

or PPC G4 Mac Mini. The stock rotational HDD is much slower, so this helps a bit.

There is a procedure to make the un-supported browser work with Intel-mac hardware.

This may or may not meet your expectation; even when working well, there are limits.

Please read Writing an effective Apple Support Communities question. We are here to help. We are all users. We are not mind-readers nor clairvoyants. We only know what you write down and tell us. Be specific and thorough. Include your hardware information and installed version of macOS.

What is your Mac model? To find the model identifier open System Profiler in the Utilities' folder. It's displayed in the panel on the right. You may also find the complete model information by selecting About This Mac from the Apple menu.