Ahead of a Super Bowl that caps a tumultuous season, a new Wall Street Journal/NBC News poll depicts a developing nightmare for the National Football League: Its core audience is losing interest rapidly, a potential threat to the league’s dominant role in American culture.

The poll shows that fans are following the sport less closely, even in key demographics, such as young men, that historically propelled the league’s growth. At the same time, parents increasingly want their children to turn away from football amid growing...