Filaments in a planar chemical garden grow following tortuous, erratic paths. We show from statistical mechanics that this scaling results from a self-organized dispersion mechanism. Effective diffusivities as high as 10-5 m2 s-1 are measured in 2D laboratory experiments. This efficient transport is four orders of magnitude larger than molecular diffusion in a liquid, and ensures widespread contact and exchange between fluids in the chemical-garden structure and its surrounding environment.