Which engine type requires the shortest take-off run: a turboprop or a turbofan?
Even with similar weight, turboprop and turbofan aircraft will have different design characteristics. A turboprop will fly at lower altitudes and airspeeds, resulting in different aerodynamic designs. A turboprop also burns less fuel per hour, so it will need less fuel capacity.

That being said, we can look at some aircraft to get an idea of what the trends might be. The best example is probably the Dornier 328. It's a turboprop aircraft that seats about 30 passengers. There also happens to be a version of the plane with jet engines. This is probably about as similar as you can get in both weight and aerodynamics. According to Skybrary, the figures are as follows:

Dornier 328
MTOW 13990 kg
Distance 1000 m

Dornier 328JET
MTOW 15200 kg
Distance 1382 m

You can see the the jet derivative is a bit heavier but requires a much longer takeoff distance.

The trend is similar between other aircraft.

Dash 8-300 Turboprop
MTOW 18600 kg
Distance 1085 m

ERJ-135 Jet
MTOW 20000 kg
Distance 1580 m

CRJ-200 Jet
MTOW 21523 kg
Distance 1527 m