Aims. Ultraviolet (UV) lines of neutral carbon observed in absorption in the local diffuse interstellar medium (ISM) have long revealed that a substantial fraction of the mass of the gas lies at a thermal pressure one to three orders of magnitude above that of the bulk of the ISM. In this paper, we propose that this enigmatic component originates from shocks propagating at intermediate (V_S > 30 km s-1) and high velocities (V_S >= 100 km s-1) in the Warm Neutral Medium (WNM).Methods. Shock waves irradiated by the standard interstellar radiation field (ISRF) are modelled using the Paris-Durham shock code designed to follow the dynamical, thermal, and chemical evolutions of shocks with velocities up to 500 km s-1. Each observed line of sight is decomposed into a high pressure and a low pressure components. The column density of carbon at high pressure is confronted to the model predictions to derive the number of shocks along the line of sight and their total dissipation rate.Results. Phase transition shocks spontaneously lead to the presence of high pressure gas in the diffuse ISM and are found to naturally produce neutral carbon with excitation conditions and linewidths in remarkable agreement with the observations. The amounts of neutral carbon at high pressure detected over a sample of 89 lines of sight imply a dissipation rate of mechanical energy with a median of ~ 3x10^{-25} erg cm-3 s-1 and a dispersion of about a factor of three. This distribution of the dissipation rate weakly depends on the detailed characteristics of shocks as long as they propagate at velocities between 30 and 200 kms s-1 in a medium with a preshock density > 0.3 cm-3 and a transverse magnetic field <= 3 microGauss. We not only show that this solution is consistent with a scenario of shocks driven by supernovae remnants (SNR) but also that this scenario is, in fact, unavoidable. Any line of sight in the observational sample is bound to intercept SNRs, mostly distributed in the spiral arms of the Milky Way, and expanding in the diffuse ionized and neutral phases of the Galaxy. Surprisingly, the range of dissipation rate derived here, in events that probably drive turbulence in the WNM, is found to be comparable to the distribution of the kinetic energy transfer rate of the turbulent cascade derived from the observations of CO in the Cold Neutral Medium (CNM).Conclusions. This work reveals a possible direct tracer of the mechanisms by which mechanical energy is injected in the ISM. It also suggests that a still unknown connection exists between the amount of energy dissipated during the injection process in the WNM and that used to feed interstellar turbulence and the turbulent cascade observed in the CNM.