Enhanced lateral growth of semiconductor thin films on glass was investigated. The unidirectional growth of silicon (Si) was achieved by introducing continuous-wave laser lateral crystallization. Alloying germanium (Ge) with Si led to highly oriented lateral growth (HOLG) of SiGe thin films. The HOLG and Ge segregation in these films were explained by the constitutional undercooling model. Quasi-single crystals were grown when the Ge mole fraction was 0.3. Using a single scan of the laser on SiGe causes the amorphous phase to become crystal, which tends to decrease the stability of growth. This growth stability was improved by performing multiple scans, where the growth was dominated by regrowth from crystal to crystal.