We analyze nucleation-controlled nanocrystal growth in a solution containing surface-binding molecular ligands, which can also nucleate compact layers on the crystal surfaces. We show that, if the critical nucleus size for ligands is larger and the nucleation barrier is lower than those for crystal atoms, the ligands nucleate faster than the atoms on relatively wide crystal facets but much slower, if at all, on narrow facets. Such competitive nucleation of ligands and atoms results in ligands covering predominantly wider facets, thus excluding them from the growth process, and acts as a selection mechanism for the growth of crystals with narrower facets, the so-called nanoplatelets. The theory is confirmed by Monte Carlo simulations and validated experimentally for CsPbBr3 nanoplatelets grown from solution. We find that the anisotropic crystal growth is controlled by the growth temperature and the strength of surface bonding for the passivating molecular ligands.