The deal between popular sports-media publisher Barstool Sports Inc. and a casino operator is the latest sign of the seismic change under way in sports betting since the Supreme Court moved two years ago to legalize it widely.

U.S. casino operators, fantasy apps and betting brands from Europe and Australia are in a race for American customers after the U.S. Supreme Court in 2018 cleared the way for states outside Nevada to establish sports wagering. Twenty states and the District of Columbia have legalized it; some limit...