Using Brownian-hydrodynamic and lattice-Boltzmann simulations, we study the nonlinear response of grafted semiflexible polymers to shear flow as a function of shear rate, grafting density, and chain stiffness. Simulation results for brush height and flow stagnation layer height agree well with a mean-field theory that incorporates the interplay of hydrodynamic screening and drag-induced polymer deformation. Our predictions for the stagnation height show excellent agreement with recent experiments on the nonlinear hydrodynamic drag of DNA-grafted colloids held, in a laser trap.