Astrophysical shocks are known to accelerate charged particles to relativistic energies, making them plausible sites for cosmic ray production. Numerical simulations are widely used to study shock acceleration, particularly due to its nonlinear nature. However, they do not always reproduce efficient particle energization. These simulations typically assume a homogeneous upstream medium, and our study aims to reveal whether pre-existing turbulence may have an effect. Using a novel simulation framework, we perform fully kinetic simulations of shocks propagating in turbulent media. Our results show that at nonrelativistic high-Mach-number oblique shocks, which model conditions in supernova remnants, electron acceleration is more efficient in the presence of pre-existing compressible turbulence. Furthermore, we have extended our investigations to relativistic shocks, that can be found in powerful extragalactic sources such as active galactic nuclei. We discuss the latest results of our relativistic shock simulations with pre-existing turbulence driven by proton cosmic rays.