how do I update my safari browser?

To upgrade Safari you would need to upgrade your Mac to a later macOS version.

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. One option is Chrome which supports macOS 10.15 and later: https://www.google.com/chrome/

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

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

If you do "About this Mac..." from your Apple menu and tell us your iMac's SUB-MODEL, we can let you know if you have an upgrade path. You should—any Mac currently running OS 10.14 Mojave can be upgraded to at least 10.15 Catalina.

Here is how the "About.." window looks in macOS 10.14, showing the location of the important sub-model info:

For security reasons, please do NOT post your serial number here. The moderators will redact it and we are back to Square 1.

Also note that when you leave 10.14, older programs like Office 2011 and older, or old versions of Adobe Creative Suite will be disabled. Non-supported apps can be quickly intetfied with this frre tool:

https://www.stclairsoft.com/Go64/index.html

If you have programs that cannot be upgraded ot the new 64-bit standard and that prevents you from upgradeping, consider using the ESR version of FireFox. It is a safe aand reliable browser recognized by most sites. The Firefox download page is here: FireFox

The version to select is shown here:

Safari versions are tied to the macOS version. E.g. on macOS Mojave 10.14.6, you should have Safari 14.1.2, and that was released in September 2021. If sites want newer versions of browsers, and you can’t update macOS on that computer, then consider Firefox or Chrome as alternative browsers. Those browser developers will also have decided on a cutoff for older system software versions, but generally their supported browsers are newer and cover a few more versions system software than Apple does with Safari. Note that old versions macOS do not get security updates anymore, which makes safe browsing more difficult.

Firefox 115.4.0 ESR: macOS 10.12–10.14,

Firefox 119.0 (current): macOS 10.15 or later,

Chrome 103: macOS 10.11–10.12,

Chrome 116: macOS 10.13–10.14,

Chrome 118 (current): macOS 10.15 or later.