HOW TO UPDATE MY BROWSER, I HAVE 13.1 HIGH SIERRA?

To upgrade Safari you would need to upgrade your macOS.

If you can’t or don’t want to upgrade, you may want to use a 3rd party internet browser that supports running their current version on your Mac.

The model year of an iMac is available at Apple menu > About This Mac.

A late 2009 through mid 2011 iMac can be upgraded to High Sierra 10.13.6 but no later version. See: How to download and install macOS - Apple Support

A late 2012 through late 2013 iMac can be upgraded to Catalina 10.15.7 but no later. See: How to download and install macOS - Apple Support

A mid 2014 and late 2014 iMac can be upgraded to the latest version of Big Sur macOS 11 but no later. See: How to download and install macOS - Apple Support

A late 2015 iMac can be upgraded to the latest version of Monterey macOS 12 but no later. See: How to download and install macOS - Apple Support

A 2017 iMac can be upgraded to the latest version of Ventura macOS 13 but no later. See: How to download and install macOS - Apple Support  

A 2019 or later iMac can be upgraded to the current version of Sonoma macOS 14. See:How to upgrade to macOS Sonoma - Apple Support

According to the following article, an extended support release of Firefox appears to be available for macOS 10.13 High Sierra until September 2024, so it may be worth checking whether this possibly could be a temporary solution in this case.

https://support.mozilla.org/en-US/kb/firefox-users-macos-1012-1013-1014-moving-to-extended-support