High‐speed compressible turbulent flows typically contain discontinuities and have been widely modelled using Weighted Essentially Non‐Oscillatory (WENO) schemes due to their high‐order accuracy and sharp shock capturing capability. However, such schemes may damp the small scales of turbulence, and result in inaccurate solutions in the context of turbulence‐resolving simulations. In this connection, the recently‐developed Targeted Essentially Non‐Oscillatory (TENO) schemes, including adaptive variants, may offer significant improvements. The present study aims to quantify the potential of these new schemes for a fully‐turbulent supersonic flow. Specifically, DNS of a compressible turbulent channel flow with M = 1: 5 and Re τ = 222 is conducted using OpenSBLI, a high‐order finite difference CFD framework. This flow configuration is chosen to decouple the effect of flow discontinuities and turbulence and focus on the capability of the aforementioned high‐order schemes to resolve turbulent structures. The effect of the spatial resolution in different directions and coarse grid implicit LES are also evaluated against theWALE LES model. The TENO schemes are found to exhibit significant performance improvements over the WENO schemes in terms of the accuracy of the statistics and the resolution of the three‐dimensional vortical structures. The 6th order adaptive TENO scheme is found to produce comparable results to those obtained with non‐dissipative 4th and 6th order central schemes and reference data obtained with spectral methods. Although the most computationally expensive scheme, it is shown that this adaptive scheme can produce satisfactory results if used as an implicit LES model.