[baldlobo]

they are more efficient because it's straight through
induction(single stage centrifugal compressor on early models; multiple stage axial compressor on newer models)
combustion(flame canisters)
exhaust(single stage centrifugal turbine on early models; 2 max axial turbines)

1-2 rotating parts

vs.

multiple rotating/moving parts that use energy in an otto cycle engine


we don't use them because they are fuel inefficient; think keeping a flame thrower going for 100km(they use pounds per hour for flight)

on the plus side that engine you have posted with the motorcycle behind it is an alison 250 turbo shaft (15 stage axial compressor with a centrifugal at the end and 2 turbines, one runs the compressor the other runs the gearbox (900 shaft hp;something like 1000hp goes to just running the engine)