Pinning of superfluid vortices to magnetic flux tubes in the outer core of a neutron star supports a velocity difference of ∼105 cm s−1 between the neutron superfluid and the proton–electron fluid as the star spins down. Under the Magnus force that arises on the vortex array, vortices undergo vortex creep through thermal activation or quantum tunnelling. We examine the hydrodynamic stability of this situation. Vortex creep introduces two low-frequency modes, one of which is unstable above a critical wavenumber for any non-zero flow velocity of the neutron superfluid with respect to the charged fluid. For typical pinning parameters of the outer core, the superfluid flow is unstable over wavelengths λ≲ 10 m and over time-scales of ∼(λ/1 m)1/2 yr down to ∼1 d. The vortex lattice could degenerate into a tangle, and the superfluid flow would become turbulent. We suggest that superfluid turbulence could be responsible for the red timing noise seen in many neutron stars, and find a predicted spectrum that is generally consistent with observations.