Vortex shedding in the turbulent wake of a square cylinder is modeled by a minimal generalized empirical Galerkin model (GM). Pressure measurements on the sides and on the back of the bluff body are used as a system output to rebuild the system full state. Despite a large number of unresolved states and the intermittent character of the vortex shedding, the use of an observer makes possible the estimation of the shift mode and the growth rate of the dynamic model truncated at order two. A high gain observer is synthesized and both simulation and experimental results show promising applications for the feedback control of the vortex shedding in order to decrease the drag coefficient, even in the case of a turbulent wake flow at high Reynolds numbers.