Getting a website to load with any browser that can run in OS X 10.5 "Leopard" is going to be a challenge.
First, upgrade that Mac to OS X 10.6.3 "Snow Leopard". It will offer nothing but improvements over OS X 10.5.
To do that you will need to find a retail, white Mac OS X Snow Leopard DVD from a reputable aftermarket source. It must be in a white box bearing an image of a snow leopard:

Apple no longer sells them. No authorized download sources exist, so if you find a site offering it to download it is a scam.
Then, update OS X using Software Update. That gives you the App Store. Then, update to the latest version your Mac can run through the App Store. Refer to How to download and install macOS - Apple Support.
Then, consider installing Brave, or Firefox. One or both may be able to run in the OS X / macOS version you are able to install on that Mac.
https://brave.com/
https://www.mozilla.org/en-US/firefox/new/
Short of buying a new Mac, those are your options. Please write back if you need help.