We report 13 high-precision light curves of eight transits of the exoplanet WASP-52b, obtained by using four medium-class telescopes, through different filters, and adopting the defocussing technique. One transit was recorded simultaneously from two different observatories and another one from the same site but with two different instruments, including a multi-band camera. Anomalies were clearly detected in five light curves and modelled as starspots occulted by the planet during the transit events. We fitted the clean light curves with the jktebop code, and those with the anomalies with the prism+gemc codes in order to simultaneously model the photometric parameters of the transits and the position, size and contrast of each starspot. We used these new light curves and some from the literature to revise the physical properties of the WASP-52 system. Starspots with similar characteristics were detected in four transits over a period of 43 days. In the hypothesis that we are dealing with the same starspot, periodically occulted by the transiting planet, we estimated the projected orbital obliquity of WASP-52b to be lambda = 3.8 \pm 8.4 degree. We also determined the true orbital obliquity, psi = 20 \pm 50 degree, which is, although very uncertain, the first measurement of psi purely from starspot crossings. We finally assembled an optical transmission spectrum of the planet and searched for variations of its radius as a function of wavelength. Our analysis suggests a flat transmission spectrum within the experimental uncertainties.