Large-eddy simulations (LES) combined with Partially Stirred Reactor (PaSR) combustion model are employed to investigate the turbulent mixing, combustion mode and flame stability of a sonic hydrogen jet injecting into high-enthalpy supersonic crossflow at three momentum flux ratios J, i.e. 0.71, 2.11 and 4.00, respectively. The LES accuracy in terms of the turbulent kinetic energy, power spectra density and sub-grid Damköhler number is carefully addressed against various LES resolution criteria and the experimental mean pressure distribution on the upper wall. The ignition processes with auto-ignition and shock compression effects are identified and analyzed. At J = 0.71, the shock-induced ignition occurs behind the reflected shock wave and the combustion heat release is dominated by the premixed combustion. While for the high jet to cross-flow momentum flux ratios, e.g., J = 2.11 and 4.00, ignition happens toward around the jet orifice due to the strong bow shock and reflected shock effects and the combustion heat releases are dominated by the non-premixed combustion. Furthermore, the mechanisms of flame stabilization, local extinction and re-ignition in the transverse jet combustion in supersonic cross-flow are further analyzed with the chemical explosive mode analysis (CEMA).