A year after U.S. authorities closed Backpage.com, the biggest player in the online sex-for-sale industry, investigators are focused on three websites now dominant in the American market and the Swiss businessman who they believe may control them.

U.S. authorities, including the Department of Homeland Security and the Justice Department, are investigating whether the sites have engaged in or knowingly enabled human trafficking, prostitution and money-laundering, and what ties they may have with David Azzato, according to people...