Natures most powerful high-energy sources are capable of accelerating particles to high energy and radiate it away on extremely short timescales, even shorter than the light crossing time of the system. It is yet unclear what physical processes can produce such an efficient acceleration, despite the copious radiative losses. By means of radiative particle-in-cell simulations, we show that magnetically dominated turbulence in pair plasmas subject to strong synchrotron cooling generates a nonthermal particle spectrum with a hard power-law range (slope $p sim 1$) within a few eddy turnover times. Low pitch-angle particles can significantly exceed the nominal radiation-reaction limit, before abruptly cooling down. The particle spectrum becomes even harder ($p < 1$) over time owing to particle cooling with an energy-dependent pitch-angle anisotropy. The resulting synchrotron spectrum is hard ($ u F_ u propto u^s$ with $s sim 1$). Our findings have important implications for understanding the nonthermal emission from high-energy astrophysical sources, most notably the prompt phase of gamma-ray bursts and gamma-ray flares from the Crab nebula.
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