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Non-Newtonian fluids have a viscosity that varies with applied stress. Elastoviscoplastic fluids, the elastic, viscous and plastic properties of which are interconnected in a non-trivial way, belong to this category. We have performed numerical simulations to investigate turbulence in elastoviscoplastic fluids at very high Reynolds-number values, as found in landslides and lava flows, focusing on the effect of plasticity. We find that the range of active scales in the energy spectrum reduces when increasing the fluid plasticity; when plastic effects dominate, a new scaling range emerges between the inertial range and the dissipative scales. An extended self-similarity analysis of the structure functions reveals that intermittency is present and grows with the fluid plasticity. The enhanced intermittency is caused by the non-Newtonian dissipation rate, which also exhibits an intermittent behaviour. These findings have relevance to catastrophic events in natural flows, such as landslides and lava flows, where the enhanced intermittency results in stronger extreme events, which are thus more destructive and difficult to predict.