When a strong shock propagates through a medium with internal degrees of freedom, the thermal equilibration involves the transfer of the mechanical energy imparted by the shock to the medium's internal modes. In gases, for example, this energy relaxation within the shock structure involves the re-distribution of the translational thermal energy of the molecules into rotational, vibrational, and electronic modes via inelastic collisions. In the present work, using a toy molecular model, we study the dynamics of shock waves driven through such a dissipative gas, characterized by inelastic collisions. The medium is modelled as a system of hard disks (2D) undergoing either elastic or inelastic collisions. Inelastic collisions are only allowed if the amplitude of the collision exceeds a certain activation threshold. When the medium allows finite dissipation, we find that the shock waves are unstable and form distinctive high density non-uniformities and convective rolls on their surface. The results obtained may shed light on the instabilities observed experimentally behind strong shock waves triggering ionization, vibrational relaxation and dissociation.