Old iMac running OS X El Capitan. Safari is 11.1.2. Can I update safari and how? I keep getting message that connection is not private on almost all websites.

As mentioned the Safari versions go hand in hand with the macOS versions.

If your iMac can not be updated past macOS 10.13 High Sierra w/Safari 13 to macOS 10.14 Mojave w/Safari 14 or later?

see > https://support.apple.com/en-us/HT211683

Then your best bet, is to get and use a third party Web Browser like FireFox.

see > https://www.mozilla.org/en-US/firefox/mac

Give this a try: boot into Safe Mode according to How to use safe mode on your Mac and test to see if the problem persists. Reboot normally and test again.

NOTE: Safe Mode boot can take up to 3 - 5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

Back up your data, check your applications for compatibility, and depending on which iMac model you have, click on High Sierra, Mojave, Catalina, Big Sur, or Monterey. A 27-inch iMac can be upgraded at least as far as High Sierra.

Upgrades won't show up in the updates list.

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Safari is built into Mac OS and cannot be updated independently. Please check to see if there are any updates for Mac OS on your computer, if there are please install them.

You might be able to use Firefox or (ack, spit!) Chrome.

Click on 'check your applications for compatibility' above.

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