Turbulence can significantly accelerate the growth of dust grains by accretion of molecules. For dust dynamically coupled to the gas, the growth rate scales with the square of the Mach number, which means that the growth time-scale can easily be reduced by more than an order of magnitude. The limiting time-scale is therefore rather the rate of molecular cloud formation, which means that dust production in the interstellar medium can rapidly reach the levels needed to explain the dust masses observed at high redshifts. Thus, turbulence may be the solution to the replenishment problem in models of dust evolution in high-redshift galaxies and explain the dust masses seen at z = 7-8. A simple analytic galactic dust-evolution model is presented, where grain growth nicely compensates for the expected higher rate of dust destruction by supernova shocks. This model is simpler, relies on fewer assumptions and seems to yields a better fit to data derived from observations, compared to previous models of the same type.