Twinning is an important deformation mode in lightweight Mg alloys, but the mechanisms of nucleation and growth of twins and their interactions with solutes remain largely unresolved. Here, a new model for thermally-activated, stress-driven growth of twin boundaries is presented and the role of random and segregated solutes in controlling this growth is studied analytically and using direct molecular dynamics simulations. Twin growth occurs by the thermally-activated nucleation and expansion of twin dislocation loops on a pre-existing twin boundary. Fluctuations in the local concentration of random solutes lowers the energy barrier for this process and thus facilitates twin growth. Segregation of solutes to the twin boundary strongly increases the energy barrier and suppresses twin growth. In random solid solutions at experimental strain rates, the thermally-activated nucleation process is sufficiently fast at low applied stresses so that growth of the twin is controlled by solute strengthening of the nucleated twin loop as it expands across the twin boundary. Annealing of a deformed sample leads to solute segregation and very strong pinning so that further twinning can only be accomplished by nucleation of new twins, consistent with experimental observations. The new mechanism also (i) operates for detwinning and rationalizes complex twin shapes observed in experiments and (ii) suggests a process for dynamic strain aging commonly observed in Mg alloys at elevated temperatures.