The formation of a flat and thin leaf presents a developmentally challenging problem, requiring intricate regulation of adaxial-abaxial (top-bottom) polarity. The patterning principles controlling the spatial arrangement of these domains during organ growth have remained unclear. Here we show that this regulation in Arabidopsis thaliana is achieved by an organ-autonomous Turing reaction-diffusion system centred on mobile small RNAs. The data illustrate how Turing dynamics transiently instructed by prepatterned information is sufficient to self-sustain properly oriented polarity in a dynamic, growing organ, presenting intriguing parallels to left-right patterning in the vertebrate embryo. Computational modelling demonstrates that this self-organizing system continuously adapts to coordinate the robust planar polarity of a flat leaf while affording flexibility to generate the tissue patterns of evolutionarily diverse organ shapes. Our findings identify a small-RNA-based Turing network as a dynamic regulator of organ polarity that accounts for leaf shape diversity at the level of the individual organ, plant or species.