The magnifications of compact-source lenses are extremely sensitive to the presence of low-mass dark matter haloes along the entire sightline from the source to the observer. Traditionally, the study of dark matter structure in compact-source strong gravitational lenses has been limited to radio-loud systems, as the radio emission is extended and thus unaffected by microlensing which can mimic the signal of dark matter structure. An alternate approach is to measure quasar nuclear-narrow- line emission, which is free from microlensing and present in virtually all quasar lenses. In this paper, we double the number of systems which can be used for gravitational lensing analyses by presenting measurements of narrow-line emission from a sample of eight quadruply imaged quasar lens systems, WGD J0405-3308, HS 0810+2554, RX J0911+0551, SDSS J1330+1810, PS J1606-2333, WFI 2026-4536, WFI 2033-4723, and WGD J2038-4008. We describe our updated grism spectral modelling pipeline, which we use to measure narrow-line fluxes with uncertainties of 2-10 per cent, presented here. We fit the lensed image positions with smooth mass models and demonstrate that these models fail to produce the observed distribution of image fluxes over the entire sample of lenses. Furthermore, typical deviations are larger than those expected from macromodel uncertainties. This discrepancy indicates the presence of perturbations caused by small-scale dark matter structure. The interpretation of this result in terms of dark matter models is presented in a companion paper.