Generally speaking, twelve years (or so) is about the limit for continued productive use of a Mac on the modern internet these days. After that you start to hit the security and access limitations that you are now seeing.
If the Mac cannot be upgraded to a more secure OS and/or browser, then it's time to plan for its replacement. That doesn't mean that you have to spend the money required for a new Mac. Options include lower cost refurbished Macs that are not older than say 3-5 years.
Some recommended and respected places to buy quality refurbs are:
• Refurbished Mac Deals - Apple
• New, Used and Refurbished Macs - macsales.com
• Used Macs - macofalltrades.com
Beware of buying from eBay, Craigs List, OfferUp and other similar sources. It's easy to buy a mac that has not been properly prepared for sale and will be an unusable brick.