We report on the growth of monoclinic β- and orthorhombic κ-phase Ga2O3 thin films using liquid-injection metal-organic chemical vapor deposition on highly thermally conductive 4H-SiC substrates using gallium (III) acetylacetonate or tris(2,2,6,6-tetramethyl-3,5-heptanedionato) gallium (III). Both gallium precursors produced the β phase, while only the use of the latter led to growth of κ-Ga2O3. Regardless of the used precursor, best results for β-Ga2O3 were achieved at a growth temperature of 700 °C and O2 flows in the range of 600–800 SCCM. A relatively narrow growth window was found for κ-Ga2O3, and best results were achieved for growth temperatures of 600 °C and the O2 flow of 800 SCCM. While phase-pure β-Ga2O3 was prepared, κ-Ga2O3 showed various degrees of parasitic β phase inclusions. X-ray diffraction and transmission electron microscopy confirmed a highly textured structure of β- and κ-Ga2O3 layers resulting from the presence of multiple in-plane domain orientations. Thermal conductivities of 53 nm-thick β-Ga2O3 (2.13 + 0.29/−0.51 W/m K) and 45 nm-thick κ-Ga2O3 (1.23 + 0.22/−0.26 W/m K) were determined by transient thermoreflectance and implications for device applications were assessed. Presented results suggest great potential of heterointegration of Ga2O3 and SiC for improved thermal management and reliability of future Ga2O3-based high power devices.