macOS Sierra's Browser needs an update that is unavailable.
Websites are telling me I need to update my Browser and Apple is telling me I need a new computer in order to update my Browser.
MacBook Pro 15″, macOS 10.12
Question marked as
What they are telling you is probably correct. Refer to Update to the latest version of Safari - Apple Support.
If the websites you require no longer function using the version of Safari for macOS "Sierra" and the Mac you are using cannot be updated beyond it, your options are to use another browser that may work with those sites, or purchase a newer Mac.
First try Brave, even though it may require a newer macOS version:
If Brave does not work with those websites try Firefox:
https://www.mozilla.org/en-US/firefox/new/
Same caveats apply.
Finally confirm that your Mac cannot be updated to use macOS "High Sierra" or perhaps even a newer version: How to download and install macOS - Apple Support. Scroll down to High Sierra 10.13. To my knowledge any Mac running Sierra can be updated to run High Sierra. In case you don't know, Find out which macOS your Mac is using - Apple Support.