The structure of the interface of a growing crystal with its nutrient phase largely determines the growth dynamics. We demonstrate that hematin crystals, crucial for the survival of malaria parasites, transition from faceted to rough growth interfaces at increasing thermodynamic supersaturation Δμ. Contrary to theoretical predictions and previous observations, this transition occurs at moderate values of Δμ. Moreover, surface roughness varies nonmonotonically with Δμ, and the rate constant for rough growth is slower than that resulting from nucleation and spreading of layers. We attribute these unexpected behaviors to the dynamics of step growth dominated by surface diffusion and the loss of identity of nuclei separated by less than the step width w. We put forth a general criterion for the onset of kinetic roughening using w as a critical length scale.