The structural phase transition of gadolinium-scandium-gallium garnet (Gd3Sc2Ga3O12, GSGG) has been studied at high pressure and high temperature using the synchrotron X-ray diffraction technique in a laser-heated diamond anvil cell. The GSGG garnet transformed to an orthorhombic perovskite structure at approximately 24 GPa after laser heating to 1500-2000 K. The garnet-to-perovskite phase transition is associated with an similar to 8% volume reduction and an increase in the coordination number of the Ga3+ or Sc3+ ion. The orthorhombic perovslcite GSGG has bulk modulus B-o = 194(15) GPa with B-o' = 5.3(8), exhibiting slightly less compression than the cubic garnet structure of GSGG with B-o = 157(15) GPa and B-o' = 6.5(10). Upon compression at room temperature, the cubic GSGG garnet became amorphous at similar to 65 GPa. Coupled with the amorphous-to-perovskite phase transition in Y3Fe5O12 and Gd3Ga5O12 at high-pressure-temperature conditions, we conclude that amorphization should represent a new thermodynamic state resulting from hindrance of the garnet-to-perovskite phase transition, whereas the garnet-to-amorphous transition in rare-earth garnets should be kinetically hindered at room temperature.