Quasar accretion disc winds observed via broad absorption lines (BALs) in the UV produce strong continuous absorption in X-rays. The X-ray absorber is believed to serve critically as a radiative shield to keep the outflow ionizations low enough for radiative driving. However, previous studies have shown that 'mini-BAL' and narrow absorption line (NAL) outflows have dramatically less X-ray absorption than BALs. Here, we examine X-ray and rest-frame UV spectra of eight mini-BAL quasars with outflow speeds in the range 0.1-0.2c to test the hypothesis that these extreme speeds require a strong shield. We find that the X-ray absorption is weak or moderate, with neutral-equivalent column densities N H < few × 1022 cm−2, consistent with mini-BALs at lower speeds. We use photoionization models to show that the amount of shielding consistent with our data is too weak to control the outflow ionizations and, therefore, it is not important for the acceleration. Shielding in complex geometries also seems unlikely because the alleged shield would need to extinguish the ionizing far-UV flux while avoiding detection in X-rays and the near-UV. We argue that the outflow ionizations are kept moderate, instead, by high gas densities in small clouds. If the mini-BALs form at radial distances of the order of R ∼ 2 pc from the central quasar (broadly consistent with theoretical models and with the mini-BAL variabilities observed here and in previous work), and the total column densities in the mini-BAL gas are N H 1021 cm−2, then the total radial extent of outflow clouds is only ΔR clouds 3 × 1013 cm in cases of no/weak shielding or ΔR clouds 3 × 1014 cm behind the maximum shield allowed by our data. This implies radial filling factors ΔR clouds/R 5 × 10− 6 or 5 × 10− 5 for the unshielded or maximally shielded cases, respectively. Compared to the transverse sizes 8 × 1015 cm (based on measured line depths), the outflows have shapes like thin 'pancakes' viewed face-on, or they occupy larger volumes like a spray of many dense clouds with a small volume filling factor. These results favour models with magnetic confinement in magnetic disc winds. To the extent that BALs, mini-BALs and NALs probe the same general outflow phenomenon, our result for dense substructures should apply to all three outflow types.