Please confirm what model you have.
'Apple menu > About this Mac'. Then click the 'more info' button. That will show a system report. Look of the 'Model Identifier' & post it here please.
Your post lists an iMac G5, this is only able to run older OS's because that type of processor (PowerPC, new Macs are Intel) is not supported anymore. You may be able to run 10.5 if the model is capable, but that is not the newest Mac OS (that is 10.10 - Yosemite). I'm afraid you won't be able to run the 'latest & greatest' web browser on such old hardware.
You will need to try other browsers such as Firefox, Opera or Chrome to see if they can work, you will also need to look for the PowerPC versions of these apps, otherwise they will not run.
The only other option I can think of is running a PowerPC version of Ubuntu, it may support a newer Chrome browser, but that is not simple to setup and may erase Mac OS.