The development and application of a compressible hybrid lattice Boltzmann method to high Mach number supercritical and dense gas flows are presented. Dense gases, especially in Organic Rankine Cycle turbines, exhibit non-classical phenomena that offer the possibility of enhancing turbine efficiency by reducing friction drag and boundary layer separation. The proposed numerical framework addresses the limitations of conventional lattice Boltzmann method in handling highly compressible flows by integrating a finite-volume scheme for the total energy alongside a non-ideal gas equation of state supplemented by a transport coefficient model. Validations are performed using a shock tube and a three-dimensional Taylor-Green vortex flow. The capability to capture non-classical shock behaviors and compressible turbulence is demonstrated. Our study gives the first analysis of a turbulent Taylor-Green vortex flow in a dense Bethe-Zel'dovich-Thompson gas and provides comparisons with perfect gas flow at equivalent Mach numbers. The results highlight differences associated with dense gas effects and contribute to a broader understanding of non-ideal fluid dynamics in engineering applications.