I cannot update my MacBook Air past the Big Sur

The Age of this computer will Govern what newer version of macOS it will Qualify to install. Use About this Mac from the Apple Icon on Desktop - the required Information will be there.

With the above information in-hand - follow this How to get old versions of macOS and verify what version this computer Qualifies to install.  For Best results use Safari to commence the download as Others may not work.

Safari will open a redirect and open the Apple Store where the download can commence.

Does this computer appear as Listed and qualifying for an upGrade to a newer version of macOS? 

The below links will help identify which version of macOS  this computer may Qualify to run.

Computer Qualifying for macOS 13 Ventura 

Computer Qualify for the Monterey upgrade

macOS Big Sur 11 hardware requirements

If Not Listed as Qualifying for Monterey or Ventura

Explore the Apple Trade-In  Process where the Value of the exiting device could be used against the cost of a new Apple Computer, iPhone or iPad  

Or Shop Refurbished Apple computers at an Apple Reduced Cost

Ventura requires a 2018 or later MacBook Air. If that's what you have, what happens when you try to upgrade?

Just to add, Monterey requires an early 2015 or later MacBook Air.