I have a 2010 iMac running el Capitan. I need to update my browser. Big Sur won't operate on it. I've tried to find an older browser unsuccessfully

Big Sur won't but High Sierra will and that may give you more browser options. See the following from Identify your iMac model - Apple Support

iMac (21.5-inch, Mid 2010)

Model Identifier: iMac11,2

Part Number: MC508xx/A, MC509xx/A

Latest compatible operating system: macOS High Sierra 10.13.6

and also see: How to get old versions of macOS - Apple Support

As you may know, Safari is only installed as part of the operating system. My suggestion is to install another browser such as Firefox or Chrome. They can be individually updated. No browser will be able to open every web site, but having additional browsers at your disposal will usually resolve any access problems.

Once you have upgraded from El Capitan to High Sierra 10.13.6, you can install Firefox 91.0.2 or the current Chrome browser. The Safari browser in High Sierra is seriously outdated from current security protocols, and you may encounter issues with selected higher security websites. There is no Safari update from Apple for operating systems (presently) older than macOS Mojave (10.14).

Try Waterfox works fine and seems to be less memory and resources-hungry than Firefox

https://www.waterfox.net/