Piston engines are significantly less expensive to operate, as long as the aircraft's performance and reliability requirements can be satisfied by a piston engine.
It's not just the fuel, it's the maintenance. Turbines run longer between overhauls and rarely fail, but the overhaul costs tens of thousands, and more if repairs are required. There just aren't any cheap parts or cheap repairs to make.
Even though older carbureted designs have poor fuel economy, small turbines (under 1000 bhp) are even worse in that regard. As turbines get larger, they get more efficient, but also more expensive, even per kW, both to buy and to maintain. Modern fuel-injected piston engines are the cheapest to own, with tolerable fuel consumption (helped by supporting lead-free car gasoline) and tolerable maintenance cost. Not considering electric aircraft, which promise the lowest costs ever, but are too new to judge decisively.
The five cheapest turboprops will all cost over $100k/year just for owning them, no flying hours included.
Comparing apples to apples, this site cites USD 1100/hour for the Cessna 208, one of the cheapest turboprops to fly. Let's look at some comparable pistons and turboprops:
- The piston-powered Cessna 421C, which is actually faster than the 208, is USD 800/hour. The Cessna 425 - a turboprop version of the 421 - costs USD 1700/hour to fly.
- The piston-powered Beechcraft Queen Air 80 is USD 900/hour, while the King Air 90 turboprop (slightly faster, but the same range and capacity) is USD 1,500/hour.
- The Piston Beaver is USD 700/hour and the otherwise identical Turbine Beaver is USD 1100/hour.
That's a 50%+ premium for a small turboprop over a comparable piston engine equipped aircraft.
Turboprops simply cost more: to buy, to own, to fly. They offer better performance, reliability, and lighter engine weight. They also scale up well. After the initial premium for switching to turbines, cost per seat starts to go down.
There is a limit beyond which piston engines don't work well: they can't fly very high, they don't scale up well, and they aren't reliable enough for a common carrier airline. So there's no modern piston equivalent to commercial turboprops. But turboprops don't scale down well, since a smaller one still needs lots of expensive precision engineering.
As a result, within the size and performance envelope where piston-powered airplanes are still a thing, they are more economical. This is despite the higher cost of less energy-dense avgas. Diesels would've been the ultimate fuel-sippers, but they remain rare for now, with electrics taking up the efficiency niche.
NB: This answer applies to the kinds of aircraft that commonly fly today and where a choice between piston and turboprop engines is possible. Large aviation piston engines now survive only as historical items, like in warbirds.