An initially two-dimensional turbulent boundary layer was subjected simultaneously to a transverse shear, imposed by a moving wall, and a strong streamwise acceleration, and the joint effects were studied computationally by applying a second-moment closure model with new low Re number and wall proximity modifications. The model was previously verified in the computation of several two-dimensional thin shear flows, including the cases in which each of the two considered external effects were present separately. The comparison with the available, though modest, experimental results shows good agreement. The computations show a dominant effect of the imposed transverse shear manifested in a large turbulence production in the initial region of the three-dimensional boundary layer, but the strong acceleration subsequently becomes dominant over the flow and damps the turbulence, leading to eventual relaminarization. The dynamics of the turbulence response, which is strongly dependent on the relation between the magnitudes of the two counteracting effects, was well reproduced by the applied model.