How to stop dealing with flat tires?
You are correct that the type of puncture you're getting is caused by the rim "pinching" the inner tube. 

The root cause here is one of the following:


pressure too low (most likely) 
You're not avoiding potholes carefully enough
Weight is too high for the tires/terrain.


This has nothing to do with fixies except perhaps that people on fixies tend to awkwardly power through obstacles without unloading enough weight at the right instant in time.

There is junk you can inject into your tubes, but that stuff is terrible and increases rotating weight. You don't need it. Also, this has nothing to do with tire quality. Your rim is pinching the inner tube, not something from the outside the tire.

Suggest you increase pressure on tires and avoid potholes. If that fails to work, try fatter tires. It is OK to go beyond the pressure specified on sidewall.