Q: how do I update my power mac G5 (my version is 10.5.8) or How do I upgrate my safari 10.5.8 to a better version

So we know more about it...

At the Apple Icon at top left>About this Mac, report the version of OSX from that window, then click on More Info, then click on Hardware> and report this upto but not including the Serial#...

Hardware Overview:

Model Name: eMac

Model Identifier: PowerMac6,4

Processor Name: PowerPC G4 (1.2)

Processor Speed: 1.42 GHz

Number Of CPUs: 1

L2 Cache (per CPU): 512 KB

Memory: 2 GB

Bus Speed: 167 MHz

Boot ROM Version: 4.9.2f1

A G5 Mac can’t be upgraded past Mac OS X 10.5.8 and Safari 5.0.6.

(129838)

TenFourFox is the most up to date browser for our PPCs, they even have G4 & G5 optimized versions...

http://www.floodgap.com/software/tenfourfox/

I'd avoid vesrion 24 though, as it no longer supports plug-ins like Flash.

http://code.google.com/p/tenfourfox/downloads/list

SeaMonkey seems pretty fast also, with many options...

http://www.seamonkey-project.org/

http://www.seamonkey-project.org/releases/

Might also look into iCab & OmniWeb versions.