How to update Safari to latest version?

Safari is updated with the macOS version. If you are running macOS Ventura you have Safari 16 already.

If you are running macOS 11 Big Sur or macOS 12 Monterey, then check the App Store app for updates.

If you are running macOS 10.15 Catalina then you cannot get Safari 16.

click here ➡️ Update to the latest version of Safari - Apple Support

What macOS version is your old Mac running? Apple Menu ➜ About this Mac.

As I said, you need at least macOS 11 Big Sur to get Safari 16.

If you have Safari 13, it suggests you have macOS 10.15 Catalina, 10.14 Mojave or 10.13 High Sierra and you cannot get Safari 16 on any of those versions of macOS.

You can use Chrome sure. Other options also include Firefox and Brave which is based on Chrome but works much better in many regards.

https://brave.com/