You should be able to use third party browsers such as Google Chrome or Vivaldi. Unfortunately Firefox only supports macOS 10.12+ now.
A system with 4GB of RAM is not much any more, but unfortunately you cannot upgrade the memory in a MBAir.
You should be able to upgrade to as least macOS 10.11 El Capitan. If you have a 2010 or 2011 model, then you can upgrade to macOS 10.13 High Sierra. Unfortunately Apple will stop providing security updates and support for High Sierra very soon. If you have a 2012+ model, then you can run macOS 10.14 Mojave and 10.15 Catalina. On some Macs you may first need to upgrade to 10.11 before being able to upgrade to later versions of macOS (if your hardware is supported).
To find the exact model of your Mac click on the Apple menu and select "About this Mac". If this doesn't show your exact model, then enter your serial number here:
https://checkcoverage.apple.com/
Here is an Apple article with instructions for creating a bootable macOS USB installer plus it includes links to download the various macOS installers:
https://support.apple.com/en-us/HT201372
I highly recommend you have a good backup before attempting to upgrade macOS just in case something goes wrong. I also suggest creating the bootable macOS USB installer as well as it gives you more options if something goes wrong.