Plants develop new organs to adjust their bodies to dynamic changes in the environment. How independent organs achieve anisotropic shapes and polarities is poorly understood. To address this question, we constructed a mechano-biochemical model for Arabidopsis root meristem growth that integrates biologically plausible principles. Computer model simulations demonstrate how differential growth of neighboring tissues results in the initial symmetry-breaking leading to anisotropic root growth. Furthermore, the root growth feeds back on a polar transport network of the growth regulator auxin. Model, predictions are in close agreement with in vivo patterns of anisotropic growth, auxin distribution, and cell polarity, as well as several root phenotypes caused by chemical, mechanical, or genetic perturbations. Our study demonstrates that the combination of tissue mechanics and polar auxin transport organizes anisotropic root growth and cell polarities during organ outgrowth. Therefore, a mobile auxin signal transported through immobile cells drives polarity and growth mechanics to coordinate complex organ development.