Time-varying thrust has been measured on a rotor in shallow turbulent flow at laboratory scale. The onset flow has a turbulence intensity of 12% at mid depth and a longitudinal turbulence length scale of half the depth, about 5 times the vertical scale, typical of shallow flows. The rotor is designed to have thrust and power coefficient variations with tip speed ratio close to that of a full-scale turbine. Three extreme probability distributions give similar thrust exceedance values with the Type 1 Pareto in mid range which gives 1:100, 1:1000 and 1:10 000 exceedance thrust forces of 1.38, 1.5 and 1.59 times the mean value. With opposing waves superimposed the extreme thrust distribution has a very similar distribution to the turbulent flow only. Exceedance forces are predicted by superposition of a drag force with drag coefficient of 2.0 based on the wave particle velocity only and with an unchanged mean thrust coefficient of 0.89. These values are relevant for the design of support structures for marine turbines.