The Galactic interstellar turbulence affects the density distribution and star formation. We introduce a new method of measuring interstellar turbulent density spectra by using the dispersion measures (DMs) of a large sample of pulsars. Without the need of invoking multiple tracers, we obtain nonuniversal density spectra in the multi-phase interstellar medium over different ranges of length scales. By comparing the analytical structure function of DMs with the observationally measured one in different areas of sky, we find a shallow density spectrum arising from the supersonic turbulence in cold interstellar phases, and a Kolmogorov-like density spectrum in the diffuse warm ionized medium (WIM). Both spectra extend up to hundreds of pc. On larger scales, we for the first time identify a steep density spectrum in the diffuse WIM extending up to several kpc. Our results show that the DMs of pulsars can provide unique new information on the interstellar turbulence.
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