Growth technologies of oxide crystals in W/Mo crucibles have been developed as a low-cost alternative to conventional processes involving Ir crucibles. Carbon-containing atmosphere needed to protect crucibles from oxidation leads to the introduction of carbon into the crystal lattice and creation of carbon-related defects, which affect the scintillation performance. Meanwhile, a search for fast scintillators for the new generation of positron-emission tomographs and high-energy physics experiments at colliders is under way. Codoping with divalent cations has become an efficient way to suppress long components of scintillation decay in Ce-doped scintillators. This work addresses Y 3 Al 5 O 12 (YAG) crystals codoped with carbon, cerium activator, and divalent cations. Optical and scintillation properties of YAG:Ce,C, $\text{A}^{2+}$ crystals (A = Ca $^{2+}$ , Mg $^{2+}$ , Ba $^{2+}$ , Sr $^{2+}$ ) are systematically studied. Among all the studied garnet compounds, YAG:Ce,C,Ca $^{2+}$ crystals demonstrated the fastest scintillation decay times, which are promising for the mentioned applications. Mechanisms of scintillation process in the studied materials are discussed. The Ce $^{3+}$ /Ca $^{2+}$ ratio in YAG:Ce,C,Ca $^{2+}$ was optimized to minimize slow components in scintillation decay.