old system, old browser, old mac

Which OS you can upgrade to will be governed by which model and year your mac is.

Click on the  top left of your screen click on About This Mac to find out the year of your mac.

If you find your mac can upgrade to the newer OS's like macOS Mojave 10.14, macOS Catalina 10.15, 

or macOs Big Sur 11.6 I would recommend you upgrade to macOS High Sierra 10.13 first.

For High Sierra, Mojave or Catalina,

Open Safari and click on this link, 

How to get old versions of macOS – Apple Support

Go to Check Compatibility to see if your mac qualifies for the OS upgrade you want.

Then go to the Download OS section and click on the OS you want to install.

Before upgrading make a full back up of your Mac.

Check that third party apps you use have updates available for the new OS.

https://roaringapps.com

Check that your printers/ scanners etc have updated drivers and software available for the new OS.

Connect your mac to your router via cable rather than rely on WiFi.

Do not use your mac while it is downloading

and do not use any other devices that share the same internet connection.

If the App Store says ‘Open’ then it is detecting a previous download of the install app

go to the Applications folder locate Install macOS ******.app.

Move it to the Trash, empty the Trash.

Restart, click on the link again, the App Store should say ‘Get' or 'Download'.

If you want to or are able to install Big Sur, open Safari and click on the link below.

How to upgrade to macOS Big Sur - Apple Support

If you are considering upgrading to Catalina or Big Sur be warned they do not support 32bit apps.

Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,

some updates may be free, some may be paid. Use this free app to discover which apps you have

are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see

if they are offering 64bit updates.

As has been said many times on this forum, whether you can upgrade your operating system depends on what year and model Mac you have. And upgrading the OS also upgrades the Safari browser. You say, you are using Chrome. Are you not able to install a newer version of Chrome? If not, perhaps you can install a newer version of another browser such as Firefox. You can check it out.