  Constraining the strength of gas turbulence in protoplanetary discs is an
open problem that has relevant implications for the physics of gas accretion
and planet formation. In this work, we gauge the amount of turbulence in 6 of
the discs observed in the DSHARP programme by indirectly measuring the vertical
distribution of their dust component. We employ the differences in the gap
contrasts observed along the major and the minor axes due to projection
effects, and build a radiative transfer model to reproduce these features for
different values of the dust scale heights. We find that (a) the scale heights
that yield a better agreement with data are generally low ($\lesssim 4$ AU at a
radial distance of $100$ AU), and in almost all cases we are only able to place
upper limits on their exact values; these conclusions imply (assuming an
average Stokes number of $\approx10^{-2}$) low turbulence levels of
$\alpha_{\rm SS}\lesssim10^{-3}-10^{-4}$; (b) for the 9 other systems we
considered out of the DSHARP sample, our method yields no significant
constraints on the disc vertical structure; we conclude that this is because
these discs have either a low inclination or gaps that are not deep enough.
Based on our analysis we provide an empirical criterion to assess whether a
given disc is suitable to measure the vertical scale height.
