Aims. We demonstrate that cosmic rays suppress the thermal overstability of radiative shocks for a wide range of upstream parameters and initial conditions. Methods. We perform one-dimensional time-dependent two-fluid simulations, introducing cosmic rays into overstable radiative shocks. Results. When a cosmic ray fluid is introduced, the oscillations of the shock front are damped and a smooth cosmic ray modified shock develops and propagates away from the cold dense layer. A large increase in

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... osmic ray pressure occurs for lower Mach numbers in radiative cosmic ray modified shocks than in adiabatic cosmic ray modified shocks. The compression ratio is limited to 7 and steady solutions do not exhibit a cold dense layer like those in radiative shocks that are not modified by cosmic rays. If the diffusion length is much larger than the cooling length, the shock is nearly isothermal. The structure of such shocks obtained with the two-fluid code agree well with analytically derived solutions for strictly isothermal flow containing cosmic rays. Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.