I have an older mac and can no longer update the OS but would like to update Safari. Is this possible?

As noted above, it can't be done. The best solution is to install a second browser such as Firefox or one of the many which are available in the Apple App Store. They generally have no problems and can also be individually upgraded.

Safari in built into the version of macOS installed on the computer. Apple does not normally issue Standalone updates for Safari.

The only way to update Safari is to explore upGrading the 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

Malware Scanning Program or a piece of malware?

There are no known Windows-like Viruses in the wild that self replicate and affect macOS, because of the underling UNIX  Foundation and Permission Limitation. 

The The Built in Security  is all that is required to protect the computer.

Should “ Certain & Specific Software “ referenced from above be installed - it will negatively impact macOS. It is suggested, to download directly from a respected ASC Contributor, the application Malwarebytes for Mac. 

It is free or paid for added features. 

Disclaimer: I am not affiliated with nor receive compensation from the Developer.

Run the Application and it should remove the malware / adware. Once done, restart computer and test.

First of all, in the future, please post your own separate issues on their own, not in an existing different conversation. All of those anti virus apps , except for Malwarebytes, do cause problems, will definitely slow down your computer and should be immediately removed. The best way is to visit their web site and download an uninstaller app. It will remove all of the various bits and pieces of the app which are locate throughout the computer.

Not Safari, but there are better browsers...

Might try this, it's Chrome Browser without google & without Ads...

https://brave.com/