Direct numerical simulations of compressible turbulent channel flows are performed for bulk Mach numbers 0.8 and 1.5 and bulk Reynolds numbers in the range of 3000-34000. The compressibility effects are well accounted for when using the semilocal scaling, as expected. Compared to incompressible flows at comparable semilocal Reynolds number Reτc∗, the mean velocity profile perfectly collapses after the Trettel and Larsson transformation [Trettel and Larsson, Phys. Fluids 28, 026102 (2016)PHFLE61070-663110.1063/1.4942022]. Furthermore, the Reynolds stresses, such as their peak values and locations, follow similar trends as in incompressible cases. In particular, the Reynolds shear stress peak follows the nonuniversal scaling transition predicted by Chen et al. [J. Fluid Mech. 871, R2 (2019)10.1017/jfm.2019.309]. Although the streamwise Reynolds stress peak continuously increases with Mach number, this increase becomes smaller as the Reynolds number increases. The streamwise and spanwise velocity spectra reveal that the typical eddy size does not vary with the Mach number when scaled using the local friction velocity and thermodynamic properties. Additionally, the compressibility contribution to the skin friction continuously decreases with Reynolds numbers. These results suggest that incompressible and compressible flows differ little at sufficiently high Reynolds number.