In this study, an accurate shock- and interface-capturing method using curvilinear body-fitted structured grids is introduced to simulate compressible multiphase flows with shockwaves. A five-equation model—proficient in capturing unsteady shocks in compressible multiphase flows without nonphysical spurious oscillations—was enhanced by extending it to a multidimensional general curvilinear coordinate system. A new eigensystem was constructed for a monotonic upstream-centered scheme for conservation laws/Godunov-type finite volume scheme that can capture strong shocks in compressible multiphase flows. The developed method was validated using a typical test involving a free-field underwater explosion that produces a strong shockwave in addition to bubble interfaces. Numerical results were compared and found to agree well with the empirical equation and previously published results. Further, the interaction of a shockwave with a wedge was computed to evaluate the shockwave propagation characteristics and appropriate treatment of solid wall boundary conditions. The numerical results show good agreement with the experimental data. Finally, the impact and propagation characteristics of shockwaves in two more complex cases, involving one and two explosions near a rigid cylinder, were numerically analyzed, and the results indicate high impacts of primary blast waves on the structure.