At supercritical conditions, thermodynamics may become strongly nonlinear which is reflected by large thermodynamic Jacobian values. This means that small variations in density, momentum or energy can result in large pressure perturbations. Because of such nonlinearities, simulations of high-pressure flows are subject to stability issues when the conservative form of the Navier-Stokes system is employed. In high-Reynolds number simulations, transported quantities are altered by numerical scheme errors, stabilization methods and filtering procedures. These alterations affect density and energy independently and may lead to significant pressure oscillations due to the nonlinear behavior of the equation of state. The objective of the present work is to compare three methods based on different sets of transport equations, to measure their impact on pressure and mixing temperature. A classical fully-conservative approach based on the transport of internal energy is compared to two quasi-conservative methods: a pressure-based and an enthalpy-based formulation. A suite of test cases of increasing complexity is employed to expose the main differences between these approaches, at conditions relevant for engineering applications. The analysis first confirmed that the energy-based formulation is prone to artificial pressure fluctuations caused by the interaction between thermodynamic nonlinearities and the numerical dissipation introduced by the stabilization scheme. It is also demonstrated that despite the stability properties of the pressure-based formulation, the mixing temperature is not properly captured since the pressure equation is not used in its conservative form. The enthalpy-based approach however, offers an improved representation of the mixing thermodynamic state while avoiding non-physical pressure variations.