Aims. Radio synchrotron polarization maps of the Galaxy can be used to infer the properties of interstellar turbulence in the diffuse magneto-ionic medium (MIM). In this paper, we investigate the normalized spatial gradient of linearly polarized synchrotron emission (|∇ P|/|P|) as a tracer of turbulence, the relationship of the gradient to the sonic Mach number of the MIM, and changes in morphology of the gradient as a function of Galactic position in the southern sky. Methods. We used data from the S-band Polarization All Sky Survey (S-PASS) to image the normalized spatial gradient of the linearly polarized synchrotron emission (|∇ P|/|P|) in the entire southern sky at 2.3 GHz. The spatial gradient of linear polarization reveals rapid changes in the density and magnetic fluctuations in the MIM due to magnetic turbulence as a function of Galactic position. We made comparisons of these data to ideal MHD numerical simulations. To constrain the sonic Mach number (M s), we applied a high-order moments analysis to the observations and to the simulated diffuse, isothermal ISM with ideal magneto-hydrodynamic turbulence. Results. We find that polarization gradient maps reveal elongated structures, which we associate with turbulence in the MIM. Our analysis indicates that turbulent MIM is in a generally transonic regime. This result for the turbulent regime is more general than the ones deduced by the analysis of electron density variation data, because it is based on the stochastic imprints of the Faraday rotation effect, which is also sensitive to the magnetic field fluctuations. Filamentary structures are seen with typical widths down to the angular resolution, and the observed morphologies closely match numerical simulations and, in some cases, Hα contours. The |∇ P|/|P| intensity is found to be approximately log-normal distributed. No systematic variations in the sonic Mach number are observed as a function of Galactic coordinates, which is consistent with turbulence in the WIM, as inferred by the analysis of Hα data. We conclude that the sonic Mach number of the diffuse MIM appears to be spatially uniform towards the Galactic plane and the Sagittarius-Carina arm, but local variations induced by nearby extended objects are also found. © ESO, 2014.