how do i update safari on my mac purchased in 2013

If you're using a version older than 13.1.2, check for updates in the Mac App Store. Otherwise, no newer version of it will run on your computer; if needed, use a product such as Firefox for websites requiring a more-recent browser.

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Safari is upgraded when you upgrade the operating system on your Mac.

If you cannot upgrade the operating system on your Mac, you may want to consider using a 3rd party browser that supports running its current version with the latest version of macOS that you can install. One option is Chrome.

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 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 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 macOS - Apple Support

A late 2015 iMac can be upgraded to the latest version of Monterey macOS 12. See:How to download macOS - Apple Support

A 2017 or later iMac can be upgraded to the current version of Ventura macOS 13. See: How to upgrade to macOS Ventura - Apple Support

Be aware that if you upgrade to Catalina or later you will also need to update any non-Apple 32-bit apps if you have any. See: https://www.macworld.com/article/232735/how-to-check-if-mac-software-is-32-or-64-bit.html