Why don't small planes use turbine (turboprop) engines?
First, piston engines are more efficient than turboprops, so their operational cost are lower. This also means that the system mass (engine plus fuel) for a trip is lower once you go beyond small ranges. In a helicopter, the engine mass is more important, because average flight times are much shorter, so you find many helicopters with turbo engines and only few with pistons.

Next, there is a well developed infrastructure for maintenance on those pistons, and new turboprops would need new, expensive infrastructure (trained people, tools, spare parts ...)

And then there is simply no incentive to develop a new GA engine. The expense to get it certified is too large for the rather small market. This is the same as for Diesel engines.