Pulsed field magnetization (PFM) of a high-J_c MgB_2 bulk disk has been investigated at 20 K, in which flux jumps frequently occur for high pulsed fields. Using a numerical simulation of the PFM procedure, we estimated the time dependence of the local magnetic field and temperature during PFM. We analyzed the electromagnetic and thermal instability of the high-J_c MgB_2 bulk to avoid flux jumps using the time dependence of the critical thickness, d_c(t), which shows the upper safety thickness to stabilize the superconductor magnetically, and the minimum propagation zone length, l_m(t), to obtain dynamical stability. The values of d_c(t) and l_m(t) change along the thermally-stabilized direction with increasing temperature below the critical temperature, T_c. However, the flux jump can be qualitatively understood by the local temperature, T(t), which exceeds T_c in the bulk. Finally, possible solutions to avoid flux jumps in high-J_c MgB_2 bulks are discussed.