A quick web search made Shazad Mohammed the target of a months-long marketing frenzy.

Back in September, the 45-year-old systems administrator in Virginia Gardens, Fla., was uninsured due to a recent job change. He typed “private health insurance” into a search engine and landed on a site that promised to help him find coverage if he entered a few basic details. Within minutes, he says, his phone started ringing. And ringing. And ringing.

The…