One of the most popular high performance light bush planes at the moment: https://en.wikipedia.org/wiki/CubCrafters_Carbon_Cub_EX
Turbines:
The money makers: https://en.wikipedia.org/wiki/De_Havilland_Canada_DHC-3_Otte... (plus it's twin cousin)
https://en.wikipedia.org/wiki/Sherpa_Aircraft_Sherpa
So these aren't bush planes?
Edit: further, there's nothing inherent to carbon or turbines that make them unsuitable for the backcountry. Turbines are generally more reliable and have fewer moving parts. There's nothing on them that you can't fix with bailing wire that you can on a piston engine. The Wilga is overwhelmingly aluminum monocoque but if you need to fix the cowling (the main carbon part) some speed tape will do as it's not really load bearing. The cessna I took my checkride in was probably missing half it's cowling bolts :/
Surely a prop is better suited for this? You could even carry a spare "just in case".
Small/medium sized turbines are used all the time on helicopters, which routinely see backwoods operation.
Also, a bush plane is, by definition, any logistics aircraft that doesn't need a runway which those very much fall into.
In airline service engines are monitored and inspected and overhauls are done when there is evidence that it is needed. One CFM-56 on a 737 went over 40,000 hours before being removed from the airplane for rebuild.
The extreme reliability of turbine engines is why they permit twin engine airliners to operate on routes that could require up to 370 minutes of single engine operation. That is, the FAA considers the risk of a second engine failure is acceptable for over six hours. [0]