In this paper, we derive the improved regularity for two-dimensional nonlinear time fractional telegraph equations by virtue of the technic of decomposition at first. Then, the famous L2-1 formula is adopted to approximate the Caputo derivative and the central finite difference method is used for spatial discretization. The convergence accuracy of the proposed method is second order in both temporal and spatial direction. Meanwhile, for the sake of reducing the time of solving the high dimensional nonlinear problems, an efficient time two-grid algorithm is proposed. Furthermore, stability analysis of the proposed scheme is studied by the energy method. At last, numerical experiments are presented to verify the validity of the theoretical statements.