Understanding the combustion dynamics in a liquid-rocket thrust chamber requires a comprehensive knowledge of the injection and mixing mechanisms at high pressures. To characterize cryogenic fluid injection and mixing process at supercritical conditions, we establish a unified model that accommodates full conservation laws and real-fluid thermodynamics and transport phenomena over the entire thermodynamic regime of concerned. The salient features of the temporal and spatial evolution of cryogenic nitrogen jets at supercritical conditions are well captured. As a result of turbulent mixing and fluid thermodynamics, the flowfield exhibits a string of large density-gradient regimes that amplifies the axial turbulent fluctuation but damps the radial one. The baroclinic torque and volume dilatation arising from the density stratification between the jet and ambient gas play an important role in determining the flow evolution. The temperature of the nitrogen fluid increases slowly along the jet centerline when the ambient pressure approaches the critical pressure of nitrogen, due to the near critical slowdown phenomena. In addition, an increase in the ambient pressure results in an earlier transition of the jet.