SHANGHAI—Family and friends rushed to hospitals Thursday looking for loved ones among the dozens of people who died in a New Year’s Eve stampede, with some asking whether city officials had provided adequate security on a night when thousands gathered to greet 2015.

Online video and photos of the event—an annual tradition in Shanghai—showed what appeared to be thousands of people in the glittery Bund district, along the banks of the Huangpu River. There were shouts and then screams as the crowd surged and people fell underfoot,...