  The first estimation of the energy cascade rate ${|\epsilon_C|}$ of
magnetosheath turbulence is obtained using the CLUSTER and THEMIS spacecraft
data and an exact law of compressible isothermal magnetohydrodynamics
turbulence. ${|\epsilon_C|}$ is found to be of the order of ${10{^{-13}}
J.m^{3}.s^{-1}}$, 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
Alfv\'enic or 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 are
evidenced and relate ${|\epsilon_C|}$ to the turbulent Mach number and the
internal energy. These new finding have potential applications in distant
astrophysical plasmas that are not accessible to in situ measurements.
