Quasicrystal growth remains an unsolved problem in condensed matter. The dynamics of the process is studied by means of synchrotron live imaging all along the solidification of icosahedral AlPdMn quasicrystals. The lateral motion of ledges driving faceted growth at the solid-melt interface is conclusively shown. When the solidification rate is increased, nucleation and free growth of new faceted grains occur in the melt due to the significant interface recoil induced by slow attachment kinetics. The detailed analysis of the evolution of these grains reveals the crucial role of aluminum rejection, both in the poisoning of their growth and driving fluid flow.