We perform a series of two-dimensional hydrodynamic simulations of the magnetorotational collapse of a supernova core. We employ a realistic equation of state and take into account electron capture and neutrino transport by the so-called leakage scheme. Recent stellar evolution calculations imply that the magnetic fields of the toroidal components are much stronger than the poloidal ones at the presupernova stage. In this study we systematically investigate the effects of the toroidal magnetic fields on the anisotropic neutrino radiation and convection. Our results show that the shapes of the shock wave and the neutrino spheres generally become more oblate for the models whose profiles of rotation and the magnetic field are shell type and become, in contrast, more prolate for the models whose profiles of rotation and the magnetic field are cylindrical than for the corresponding models without the magnetic fields. Furthermore, we find that magnetorotational instability induced by nonaxisymmetric perturbations is expected to develop within the prompt-shock timescale. Combined with the anisotropic neutrino radiation, which heats matter near the rotational axis preferentially, the growth of the instability may enhance the heating near the axis. This might suggest that magnetar formation is accompanied by a jetlike explosion.