The interstellar medium of the Milky Way is multiphase(1), magnetized(2) and turbulent(3). Turbulence in the interstellar medium produces a global cascade of random gas motions, spanning scales ranging from 100 parsecs to 1,000 kilometres (ref. 4). Fundamental parameters of interstellar turbulence such as the sonic Mach number (the speed of sound) have been difficult to determine, because observations have lacked the sensitivity and resolution to image the small-scale structure associated with turbulent motion(5-7). Observations of linear polarization and Faraday rotation in radio emission from the Milky Way have identified unusual polarized structures that often have no counterparts in the total radiation intensity or at other wavelengths(8-12), and whose physical significance has been unclear(13-15). Here we report that the gradient of the Stokes vector (Q, U), where Q and U are parameters describing the polarization state of radiation, provides an image of magnetized turbulence in diffuse, ionized gas, manifested as a complex filamentary web of discontinuities in gas density and magnetic field. Through comparison with simulations, we demonstrate that turbulence in the warm, ionized medium has a relatively low sonic Mach number, M(s) less than or similar to 2. The development of statistical tools for the analysis of polarization gradients will allow accurate determinations of the Mach number, Reynolds number and magnetic field strength in interstellar turbulence over a wide range of conditions.