We discuss new photometry from high-resolution images of seven intermediate-age (1-2 Gyr) star clusters in the Large Magellanic Cloud taken with the Advanced Camera for Surveys on board the Hubble Space Telescope. We fit color-magnitude diagrams (CMDs) with several different sets of theoretical isochrones and determine systematic uncertainties for population parameters when derived using any one set of isochrones. The cluster CMDs show several interesting features, including extended main-sequence turnoff (MSTO) regions, narrow red giant branches, and clear sequences of unresolved binary stars. We show that the extended MSTOs are not caused by photometric uncertainties, contamination by field stars, or the presence of binary stars. Enhanced helium abundances in a fraction of cluster stars are also ruled out as the reason for the extended MSTOs. Quantitative comparisons with simulations indicate that the MSTO regions are better described by a spread in ages than by a bimodal age distribution, although we cannot formally rule out the latter for the three lowest-mass clusters in our sample (which have masses lower than similar to 3 x 10(4) M(circle dot)). This conclusion differs from that of some previous works which suggested that the age distribution in massive clusters in our sample is bimodal. This suggests that any secondary star formation occurred in an extended fashion rather than through short bursts. We discuss these results in the context of the nature of multiple stellar populations in star clusters. ; peer reviewed: yes ; NRC Pub: yes