Doesn't look like firefox will work on it either but they give some good advice on other options
https://support.mozilla dot org/en-US/kb/firefox-no-longer-works-mac-os-x-10-5
(replace the word dot with a .)
"Apple stopped supporting Mac OS X 10.5 in 2009 and many software makers have either dropped support for it or soon will. Newer versions of Firefox require OS X 10.6 (Snow Leopard) or above. Depending on the age of your computer, you may be able to upgrade to a newer version of OS X. If you can't upgrade to OS X 10.10 (Yosemite), newer versions of Firefox will also work on 10.9 (Mavericks), 10.8 (Mountain Lion), 10.7 (Lion) and 10.6 (Snow Leopard). Apple still supports these older operating systems, although it's likely that it will drop support for 10.6 (Snow Leopard) soon."
It is possible to download older versions of browsers and stop them from trying to update themselves but I wouldn't recommend it since you'd be missing out on a lot of security updates.
Do you have a smartphone? Otherwise, as you say, maybe time for a new lappy.