Hello. It sounds like you have an older MacBook Air that is unable to upgrade beyond macOS 10.13. Assuming this is the case, you will not be able to upgrade Safari on this computer as no further Safari updates are available for macOS 10.13. Your best bet for continuing to browse the web safely is to use an alternate browser such as Google Chrome (as you mentioned). If you want to upgrade beyond macOS 10.13 and upgrade Safari, you will need to upgrade to a newer Mac.
A list of computers that can run the latest version of macOS, Big Sur, is available here.