Context. Strong lenses are extremely useful probes of the distribution of matter on galaxy and cluster scales at cosmological distances, however, they are rare and difficult to find. The number of currently known lenses is on the order of 1000. Aims. The aim of this study is to use crowdsourcing to carry out a lens search targeting massive galaxies selected from over 442 square degrees of photometric data from the Hyper Suprime-Cam (HSC) survey. Methods. Based on the S16A internal data release of the HSC survey, we chose a sample of ∼300 000 galaxies with photometric redshifts in the range of 0.2 <  z phot  < 1.2 and photometrically inferred stellar masses of log M *  > 11.2. We crowdsourced lens finding on this sample of galaxies on the Zooniverse platform as part of the Space Warps project. The sample was complemented by a large set of simulated lenses and visually selected non-lenses for training purposes. Nearly 6000 citizen volunteers participated in the experiment. In parallel, we used Y ATTA L ENS , an automated lens-finding algorithm, to look for lenses in the same sample of galaxies. Results. Based on a statistical analysis of classification data from the volunteers, we selected a sample of the most promising ∼1500 candidates, which we then visually inspected: half of them turned out to be possible (grade C) lenses or better. By including lenses found by Y ATTA L ENS or serendipitously noticed in the discussion section of the Space Warps website, we were able to find 14 definite lenses (grade A), 129 probable lenses (grade B), and 581 possible lenses. Y ATTA L ENS found half the number of lenses that were discovered via crowdsourcing. Conclusions. Crowdsourcing is able to produce samples of lens candidates with high completeness, when multiple images are clearly detected, and with higher purity compared to the currently available automated algorithms. A hybrid approach, in which the visual inspection of samples of lens candidates pre-selected by discovery algorithms or coupled to machine learning is crowdsourced, will be a viable option for lens finding in the 2020s, with forthcoming wide-area surveys such as LSST, Euclid , and WFIRST.