Context: The discovery of multiple main sequences (MS) in the massive clusters NGC 2808 and Omega Centauri, along with multiple subgiant branches in NGC 1851 and NGC 6388 has challenged the long-held paradigm that globular clusters consist of simple stellar populations. This evolving picture has been further complicated by recent photometric studies of the Large Magellanic Cloud (LMC) intermediate-age clusters, where the main sequence turn-off (MSTO) was found to be bimodal (NGC 1806 and NGC 1846) or broadened (NGC 1783 and NGC 2173). Aims: We undertook a study of archival HST images of Large and Small Magellanic Cloud clusters with the aim of measuring the frequency of clusters with evidence of multiple or prolonged star formation events and determining their main properties. We found useful images for 53 clusters that cover a wide range of ages. In this paper, we analyse the color-magnitude diagrams (CMD) of sixteen intermediate-age (~1-3 Gyr) LMC clusters. Methods: The data were reduced and the photometry corrected for differential reddening (where required). We find that eleven clusters show an anomalous spread (or split) in colour and magnitude around the MSTO, even though the other main features of the CMD (MS, red giant branch, asymptotic giant branch) are narrow and the horizontal branch (HB) red clump well defined. By using the CMD of the stars in regions that surround the cluster, we demonstrate that the observed feature is unequivocally associated to the clusters. We use artificial-star tests to demonstrate that the spread (or split) is not an artifact due to photometric errors or binaries. Results: We confirm that two clusters (NGC 1806 and NGC 1846) clearly exhibit two distinct MSTOs and observe, for the first time, a double MSTO in NGC 1751. In these three clusters, the population corresponding to the brighter MSTO includes more than two-thirds of the cluster stellar population. We confirm the presence of multiple stellar populations in NGC 1783. Our photometry strongly suggests that the MSTO of this cluster is formed by two distinct branches. In seven clusters (ESO057-SC075, HODGE7, NGC 1852, NGC 1917, NGC 1987, NGC 2108, and NGC 2154), we observed an intrinsic broadening of the MSTO that may suggest that these clusters have experienced a prolonged period of star formation that spans a period between 150 and 250 Myr. The CMDs of IC 2146, NGC 1644, NGC 1652, NGC 1795, and NGC 1978 show no evidence of spread or bimodality within our photometric precision. In summary, 70 ± 25% of our sample are not consistent with the simple, single stellar population hypothesis. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. Table 1, Figs. 13-23, 32-36, 38, and 39 are only available in electronic form at http://www.aanda.org