The first estimation of the energy cascade rate |εC| of magnetosheath turbulence is obtained using the CLUSTER and THEMIS spacecraft data and an exact law of compressible isothermal magnetohydrodynamics turbulence. |εC | is found to be of the order of 10−13J.m−3.s−1, at least two orders of magnitude larger than its value in the solar wind (order of 10−16 J.m−3.s−1 in the fast wind). Two types of turbulence are evidenced and shown to be dominated either by incompressible Alfénic or compressible magnetosonic-like fluctuations. Density fluctuations are shown to amplify the cascade rate and its spatial anisotropy in comparison with incompressible Alfv´enic turbulence. Furthermore, for compressible magnetosonic fluctuations, large cascade rates are found to lie mostly near the linear kinetic instability of the mirror mode. New empirical power-laws relating |C | to the turbulent Mach number and to the internal energy are evidenced. These new finding have potential applications in distant astrophysical plasmas that are not accessible to in situ measurements.