We use 3D magnetohydrodynamics simulations with anisotropic thermal conduction to study turbulence due to the magnetothermal instability (MTI) in the intracluster medium (ICM) of galaxy clusters. The MTI grows on time-scales of ≲1 Gyr and is capable of driving vigorous, sustained turbulence in the outer parts of galaxy clusters if the temperature gradient is maintained in spite of the rapid thermal conduction. If this is the case, turbulence due to the MTI can provide up to 5–30 per cent of the pressure support beyond r500 in galaxy clusters, an effect that is strongest for hot, massive clusters. The turbulence driven by the MTI is generally additive to other sources of turbulence in the ICM, such as that produced by structure formation. This new source of non-thermal pressure support reduces the observed Sunyaev–Zel’dovich (SZ) signal and X-ray pressure gradient for a given cluster mass and introduces a cluster mass and temperature-gradient-dependent bias in SZ and X-ray mass estimates of clusters. This additional physics may also need to be taken into account when estimating the matter power spectrum normalization, σ8, through simulation templates from the observed amplitude of the SZ power spectrum.