Three-dimensional numerical simulations of canonical statistically steady, statistically planar turbulent flames have been used in an attempt to produce distributed burning in lean methane and hydrogen flames. Dilatation across the flame means that extremely large Karlovitz numbers are required; even at the extreme levels of turbulence studied (up to a Karlovitz number of 8767) distributed burning was only achieved in the hydrogen case. In this case, turbulence was found to broaden the reaction zone visually by around an order of magnitude, and thermodiffusive effects (typically present for lean hydrogen flames) were not observed. In the preheat zone, the species compositions differ considerably from those of one-dimensional flames based a number of different transport models (mixture averaged, unity Lewis number and a turbulent eddy viscosity model). The behaviour is a characteristic of turbulence dominating non-unity Lewis number species transport, and the distinct limit is again attributed to dilatation and its effect on the turbulence. Peak local reaction rates are found to be lower in the distributed case than in the lower Karlovitz cases but higher than in the laminar flame, which is attributed to effects that arise from the modified fuel-temperature distribution that results from turbulent mixing dominating low Lewis number thermodiffusive effects. Finally, approaches to achieve distributed burning at realisable conditions are discussed; factors that increase the likelihood of realising distributed burning are higher pressure, lower equivalence ratio, higher Lewis number and lower reactant temperature.