The most efficient IC engines are large Diesels. At the extreme end are ship engines with better than 50% thermal efficiency resulting in a specific fuel consumption of only 0.260 lbs/hp/hour or 158 g/kW-h. But even supercharged truck diesels achieve above 40% thermal efficiency at high load (this NHTSA study gives 42%).
Aerodiesels have achieved 220 g/kW-h already with the Jumo 204 and 205 of the early 1930s. Even the modern Thielert diesels (now sold by Continental) are hardly better, claiming 214 g/kW-h. Also the Napier Nomad, a super- and turbocharged aero diesel with utmost efficiency as its design goal just reached 219 g/kW-h.
Gasoline engines start at about 240 g/kW-h; this value is achieved by the Lycoming IO-390 with fuel injection. Without fuel injection, specific consumption rises to 260 - 280 g/kW-h which is typical for a Lycoming O-360 at 65% power. Note that the Jumo 213, one of the more efficient WW II-era piston engines, already achieved 260 g/kW-h even with 87 octane fuel and a compression ratio of only 6.93:1 at its most favourable operating point. Advanced Innovative Engineering, who have taken over the Norton-Wankel engine, claim 310-350 g/kW-h for their 650CS with 120 PS.
Comparing that with turboprops needs some conversion of thrust into power. This is only valid for a specific flight speed. If you do that at cruise speed, the large turboprops Progress D27 and Europrop TP400 claim a consumption of around 240 g/kW-h. Smaller turboprops rarely achieve below 300 g/kW-h.
To save you from the trouble of looking up and converting the data in the last link, here is a selected list:
- Allison 250 $\;\;\;\;\;\;\;$: 370 g/kW-h. This is a typical small helicopter engine.
- Garrett TPE331$\;\;$ : 310 g/kW-h. This is used on small turboprops like the Do-228 or the Merlin III.
- PWC 126A $\;\;\;\;\;\;\;$: 280 g/kW-h. Getting larger - BAe ATP.
- Rolls-Royce Tyne : 237 g/kW-h. This has long been the largest turboprop in the West and used on aircraft like the Canadair 400 / CL-44.
Please note that those turboprops feed on kerosene while the piston engines need gasoline. But by basing the comparison on a per-mass basis, it is valid because the energy densities of both are almost identical. Very large turboprops are as efficient as gasoline piston engines, but diesels still have a small advantage.
Now for turbofans. Here we have thrust which needs to be converted into power first by multiplying it with flight speed. It would be nonsensical to compare the static case – here by definition turbofans do not produce power. For the nitpickers: Yes, I need to look at the gas speeds ahead and behind the engine, but still, this makes for a poor comparison: Most static values are from test stands with all accessories removed and no losses for engine mounts and fairings. I will instead use the figures in cruise given in this answer, using a fuel burn of $b_f$ = 18 g/kNs and a speed of Mach 0.78, which equates to a flight speed of 262 m/s in 11.000 m altitude. Multiply by 3600 for a per-hour value and divide by 262 (the N are in the denominator!) and you arrive at 247 g/kW-h. So again, very similar to good gasoline piston engines but not as good as diesels.
But again this comparison has to be taken with the proverbial grain of salt. Now we need to have a closer look at speed. Thrust-specific consumption goes up with speed and roughly doubles between the static case and cruise speed for a modern turbofan. The GE-90 achieves 8 g/kN-s static and 15 g/kN-s at Mach 0.8 – that would be just 209 g/kW-h and be on a par with the best diesels. For comparison: The installed figures for modern military engines in supersonic aircraft are 20 g/kN-s. And regarding the fuel-hungry turbojets: The old Jumo 004 achieved 39 g/kN-s – just twice as much with a compression ratio of only 3.3:1. Real fuel guzzlers were the Argus 014 of the V-1 with 107 g/kN-s in cruise.
While turbine engines gain in efficiency with altitude due to colder intake air, the diagram below comparing the Jumo 213 A with the J version (source) shows an increase in power specific fuel consumption with altitude. Note that flight speed will also rise with altitude and is not given, so I suspect this is more due to higher speed than higher altitude. Again, these are real-world data from flight test with the engine installed in a FW-190D (source). Going from sea level to 10 km which roughly doubles true air speed raises the specific consumption by 20%.
Comparison chart between Jumo 213 A and J. Flight altitude is given in [km] along the x axis and specific consumption along the right y-axis. Multiply by 1.34 for g/kW-h. The lower set of lines are for partial load operation between 2100 and 2700 RPM (A version) rsp. 3000 RPM (J version) while the upper set of consumption lines are for maximum power operation at 3000 RPM (A version) rsp. 3400 to 3700 RPM (J version), partially with water-methanol injection.