The study of interactions of exploding and surrounding plasmas in an external magnetic field [K. Yamauchi and Y. Ohsawa, Phys. Plasmas 14, 053110 (2007)] is verified with two-dimensional (2D) electromagnetic particle simulations, for a case in which the initial velocity of the exploding plasma is perpendicular to the external magnetic field. The 2D simulations show essentially the same shock-formation processes as those in the previous one-dimensional simulation, including penetration of exploding ions into surrounding plasma, formation of a strong magnetic-field pulse due to deceleration of the exploding ions, ion reflection by the pulse, and subsequent splitting of the pulse into two magnetosonic pulses which then develop into forward and reverse shock waves. Furthermore, the 2D structure of electromagnetic fields in the region, where the exploding and surrounding ions overlap, is investigated with particular attention to the linear and nonlinear evolution of modified two-stream instabilities in the magnetic field that is being gradually compressed. The effects of these instabilities on ion reflection and on 2D magnetic fluctuations in the two generated pulses are also discussed. (C) 2013 AIP Publishing LLC.