The physics of compressible turbulence in high energy density (HED) plasmas is an unchartered experimental area. Simulations of compressible and radiative flows relevant for astrophysics rely mainly on subscale parameters. Therefore, we plan to perform turbulent hydrodynamics experiments in HED plasmas (TurboHEDP) in order to improve our understanding of such important phenomena for interest in both communities: laser plasma physics and astrophysics. We will focus on the physics of supernovae remnants which are complex structures subject to fluid instabilities such as the Rayleigh–Taylor and Kelvin–Helmholtz instabilities. The advent of megajoule laser facilities, like the National Ignition Facility and the Laser Megajoule, creates novel opportunities in laboratory astrophysics, as it provides unique platforms to study turbulent mixing flows in HED plasmas. Indeed, the physics requires accelerating targets over larger distances and longer time periods than previously achieved. In a preparatory phase, scaling from experiments at lower laser energies is used to guarantee the performance of future MJ experiments. This subscale experiments allow us to develop experimental skills and numerical tools in this new field of research, and are stepping stones to achieve our objectives on larger laser facilities. We review first in this paper recent advances in high energy density experiments devoted to laboratory astrophysics. Then we describe the necessary steps forward to commission an experimental platform devoted to turbulent hydrodynamics on a megajoule laser facility. Recent novel experimental results acquired on LULI2000, as well as supporting radiative hydrodynamics simulations, are presented. Together with the development of LiF detectors as transformative X-ray diagnostics, these preliminary results are promising on the way to achieve micrometric spatial resolution in turbulent HED physics experiments in the near future.