Updating MacBook Air 2013(later) to newer OS

Could be that the old version of Safari in Mavericks can't handle modern https:// security – such as the https:// security on the "How to download and install macOS" page!

The .DMG links in that article use plain http:// (no security), so your browser might have an easier time with them. Phil0124 supplied a link to what I presume is the El Capitan installer. This is the one for Sierra. Beyond Sierra, the links require the App Store, which is why you need to do an intermediate upgrade to something like El Capitan, or Sierra, so that you are then in a position to do a further upgrade.

http://updates-http.cdn-apple.com/2019/cert/061-39476-20191023-48f365f4-0015-4c41-9f44-39d3d2aca067/InstallOS.dmg

Back up your data, check the application compatibility table, click here, and then on the download link for a higher OS. That model's maximum one is Big Sur.

Catalina and above won't run 32-bit software; if you choose one of them, use Go64 as well.

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A Mid 2013 MacBook Air can update to macOS Big Sur, but no further. (There is no Late 2013 MacBook Air model)

click here ➜ macOS Big Sur is compatible with these computers - Apple Support

The download can be found in the following link.

click here ➜ How to download and install macOS - Apple Support

Get it from another computer or with a different web browser; if it's too new for the Mac App Store links, download and transport the El Capitan installer.

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Click here ➜ http://updates-http.cdn-apple.com/2019/cert/061-41424-20191024-218af9ec-cf50-4516-9011-228c78eda3d2/InstallMacOSX.dmg