Updating Safari using Software Update

The way Safari is updated is by upgrading your computer's software. An upgrade is when you install a newer OS version, such as upgrading from Catalina to Big Sur. An update is to install a newer version of the same operating system. The reason for this is because Safari is installed as part of the operating system. It cannot normally be individually installed. Incidentally, a newer version of Safari does not always work with every web site. Sometimes it is necessary to install another browser such as Firefox or one of the many browsers available from the Apple App Store to use with certain web sites. They are individually installed and upgraded.

So, the latest Safari version is installed when you upgrade the OS on your iMac. You can check the model year of your iMac under the Apple menu > About This Mac.

You are aware that is very obsolete hardware and software, but you can try this.

If rebooting your Mac doesn’t fix things, try applying the update while booted in Safe mode (hold the shift key down at startup). It can take much longer to safe boot (10 min) so be patient.

How to use safe mode on your Mac - Apple Support 

Safe mode disables loading third-party drivers, forces the OS to clear certain caches and logs and does other miscellaneous housekeeping. It's non-destructive to user data so is safe to do. Very often certain weird software behavior is corrected by simply logging in this way as a troubleshooting method. When you're in safe boot, the machine will not be at its best performance, especially with graphics, but that's expected.

Exit safe mode by restarting your Mac normally.

It may simply be that there is no longer a verifiable downloadable update for that very obsolete version of Safari. You might try to find a third-party browser that will work, though I think even that'll be a challenge. A browser that will run in that era Mac OS will not meet the security and privacy requirements of the modern internet and its practical use will be severely constrained.

In my humble opinion you should do yourself a favor; retire that Mac and replace it with a newer machine that can at least run Catalina and a modern browser. Find a 2015 or newer replacement.