how do update my safari browser

Safari is built into the version of macOS installed on the computer. As such, Apple does not normally offer a Standalone upgrade for Safari.

To get a newer version of Safari requires UpGrading to a newer version of macOS

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence

Tips Big Sir Upgrade Released Nov 2020

Tips Monterey Upgrade  Released Oct 25, 2021

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal  operation of the OS and may even Corrupt the OS requiring a ReinstallationTo do what is being asked - Reinstall Safari - Safari is built into the the OS. To reinstall the OS required rebooting into Recovery Mode Command + r and choose Install macOS. An active and functioning Internal Connection is Required.

To get a newer version of Safari requires UpGrading to a newer version of macOS

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence

Tips Big Sir Upgrade Released Nov 2020

Tips Monterey Upgrade  Released Oct 25, 2021

Make Time Machine Backup  is very useful and can be used to Revert to Previous Working macOS

Notation - Mojave is the Last version to support 32 bit applications / extensions and Drivers. Catalina, Big Sur & Monterey supports only 64 bit and there is no workaround.

Suggest using Go64 to find 32 bit application before attempting the upGrade. Some or many of the Application already installed may required update or upGrades to 64 Bit Versions.

Are there any AntiVirus, Disk Cleaner, Optimizers, Un-installers, etc installed which should be removed as per Developers Instructions. They are useless, unneeded, cause havoc and interfere with the normal  operation of the OS and may even Corrupt the OS requiring a Reinstallation

Safari gets updated when Apple releases updates for the version of macOS that is currently running on your Mac. It also gets updated when you install an upgrade to your OS that takes you to the next generational iteration.

To be clear, software update generally means to install software to update the macOS currently running on the Mac; update Catalina 10.15.1 to Catalina 10.15.7, for example.

To update your current macOS, click the Apple  icon > About This Mac and then click Software Updates.... If there are any updates available for your Mac, they will be listed and you can apply them when you like.  Software updates may be set to occur automatically. If no updates are listed, then it may be assumed that none are available for your current OS.

A software upgrade is usually a generational step up to a newer macOS; upgrade from 10.13 High Sierra to 10.14 Mojave, for example, or from 10.15 Catalina to macOS 11 Big Sur.

Upgrades do not happen automatically. They must be initiated by a user with admin privileges.

If there are no updates for Safari available to you, you might give a third-party browser like Brave or Firefox a try.