Fish-like robots have been widely used in intelligent surveillance and investigation because of their high swimming efficiency and low traveling noise. Numerical simulations are usually selected to simulate the movement modes and hydrodynamic characteristics of fish-like robots during design and manufacture. However, the body-fitted grid method traditionally utilized in numerical simulations often has difficulty dealing with moving solid boundaries. In this work, the immersed boundary method, superior in handling the moving boundary conditions, is employed to simulate the movement of a fish-like robot swimming in high Reynolds number flows in combination with the RANS turbulence model. The numerical method is first validated using a fluid flowing over a square block, and the corresponding results are in good agreement with the ones reported in reference. Then, the swing of the fish-like robot under three different Reynolds numbers is studied. The lift coefficient and the drag coefficient of the fish-like robot decrease with increasing the Reynolds number. This paper provides remarkable support for future designs and applications of fish-like robots.