Kevlahan et. al. have developed a dynamically adaptive wavelet method for the direct simulation of turbulent flows. Although the spatial grid adapts to the solution at each time step, a single time step is used for all positions and scales. This approach clearly does not fully exploit the spatial and temporal intermittency of turbulence. In this paper we propose a simultaneous space-time adaptive wavelet method for computation of highly intermittent turbulence. This allows the time step to vary depending on position and scale. Numerical experiments demonstrate the accuracy and efficiency of the method in 1D+t and in 2D+t, and representative results are presented.