Aims. We investigate the effects of radiative shocks on the observed X-ray emission from the type II supernova SN 1993J. To this end, the X-ray emission is modeled as a result of the interaction between the supernova ejecta and a dense circumstellar medium at an age of 8 years. Methods. The circumstances under which the reverse shock is radiative are discussed and the observed X-ray emission is analyzed using the numerical code described in Nymark et al. (2006, A&A, 449, 171). We argue that the original analysis of the X-ray observations suffered from the lack of self-consistent models for cooling shocks with high density and velocity, leading to questionable conclusions about the temperatures and elemental abundances. We reanalyze the spectra with our numerical model, and discuss the expected spectra for different explosion models for the progenitors. Results. We find that the spectra of SN 1993J are compatible with a CNO-enriched composition and that the X-ray flux is dominated by the reverse shock.