We use VLBA+VLA observations to measure the sizes of the scatter-broadened images of 6 of the most heavily scattered known pulsars: 3 within the Galactic Centre (GC) and 3 elsewhere in the inner Galactic plane. By combining the measured sizes with temporal pulse broadening data from the literature and using the thin-screen approximation, we locate the scattering medium along the line of sight to these 6 pulsars. At least two scattering screens are needed to explain the observations of the GC sample. We show that the screen inferred by previous observations of SGR J1745-2900 and Sgr A*, which must be located far from the GC, falls off in strength on scales < 0.2 degree. A second scattering component closer to (< 2 kpc) or even (tentatively) within (< 700 pc) the GC produces most or all of the temporal broadening observed in the other GC pulsars. Outside the GC, the scattering locations for all three pulsars are ~2 kpc from Earth, consistent with the distance of the Carina-Sagittarius or Scutum spiral arm. For each object the 3D scattering origin coincides with a known HII region (and in one case also a supernova remnant), suggesting that such objects preferentially cause the intense interstellar scattering seen towards the Galactic plane. We show that the HII regions should contribute > 25% of the total dispersion measure (DM) towards these pulsars, and calculate reduced DM distances. Those distances for other pulsars lying behind HII regions may be similarly overestimated.
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