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The first estimation of the energy cascade rate vertical bar epsilon(C)vertical bar of magnetosheath turbulence is obtained using the Cluster and THEMIS spacecraft data and an exact law of compressible isothermal magnetohydrodynamics turbulence. The mean value of vertical bar epsilon(C)vertical bar is found to be close to 10(-13) Jm(-3) s(-1), at least 2 orders of magnitude larger than its value in the solar wind (similar to 10(-16) Jm(-3) s(-)1 in the fast wind). Two types of turbulence are evidenced and shown to be dominated either by incompressible Alfvenic or compressible magnetosoniclike fluctuations. Density fluctuations are shown to amplify the cascade rate and its spatial anisotropy in comparison with incompressible Alfvenic 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 vertical bar epsilon(C)vertical bar to the turbulent Mach number and to the internal energy are evidenced. These new findings have potential applications in distant astrophysical plasmas that are not accessible to in situ measurements.