Most recent websites use an encryption method that Mac is not capable of decoding. By "recent" I mean websites that have been developed and supported for use with browsers for the past ten years or so. That iMac is a lot older than that, and its hardware simply cannot keep up with those more demanding browsers.
Read Avoid fraud by using encrypted websites in Safari on Mac - Apple Support for a description of those security requirements. Many websites do not require encryption, but they use it anyway. Some websites have no particular need for encryption, and don't use it. They are becoming increasingly rare.
But in the case of the specific example you cited (www.chase.com) as a financial services or banking site it would be irresponsible for them to use anything but the latest encryption.
The unfortunate fact of life today is... you need a newer Mac. If you want to save a decent amount of money on a newer one, consider perusing Apple's Refurbished Mac Deals page. You can buy a very nicely equipped iMac for about $1100 — which as I recall is about $100 less than the purchase price of yours back when it was sold.