We report, for the first time, that the energy of femtosecond optical laser pulses, E, with relativistic intensities I > 10(21) W/cm(2) is efficiently converted to X-ray radiation, which is emitted by "hot" electron component in collision-less processes and heats the solid density plasma periphery. As shown by direct high-resolution spectroscopic measurements X-ray radiation from plasma periphery exhibits unusual non-linear growth ~E(4-5) of its power. The non-linear power growth occurs far earlier than the known regime when the radiation reaction dominates particle motion (RDR). Nevertheless, the radiation is shown to dominate the kinetics of the plasma periphery, changing in this regime (now labeled RDKR) the physical picture of the laser plasma interaction. Although in the experiments reported here we demonstrated by observation of KK hollow ions that X-ray intensities in the keV range exceeds ~10(17) W/cm(2), there is no theoretical limit of the radiation power. Therefore, such powerful X-ray sources can produce and probe exotic material states with high densities and multiple inner-shell electron excitations even for higher Z elements. Femtosecond laser-produced plasmas may thus provide unique ultra-bright X-ray sources, for future studies of matter in extreme conditions, material science studies, and radiography of biological systems.