Long before the World Cup kicked off, U.S. manager Mauricio Pochettino wasn’t shy about setting extraordinarily high expectations for a country that believes footballs are oblong. And after the Americans finished atop their group, he continued to emphasize that the goal isn’t to play a game or two in the knockout rounds.

Pochettino has his sights set on taking the whole entire thing.

“Making history is winning the World Cup,” he said.

But in order to achieve global soccer supremacy, the Americans will have to go through the continent that has dogged them for nearly five years. That’s how long it has been since the U.S. managed to beat a single team from Europe.

“You’ve got to beat everyone and be able to be everyone,” said fullback Sergiño Dest. “So if it’s from Europe or Africa, it doesn’t matter. We just want to win.”

While the Americans are favored to advance to the Round of 16, what’s certain is that the team known as the Zmajevi, or dragons, won’t make the game particularly fun for their hosts. While some European nations are known for their beautiful passing or virtuosic dribbling, the Bosnians have developed a reputation for something else: turning soccer fields into boxing rings.

“We will try to create problems,” Bosnia and Herzegovina coach Sergej Barbarez said, “and of course win the game.”

None of the 32 teams that reached the knockout rounds had more fouls than Bosnia’s 46 during the group stage. And already, the team hasn’t been unapologetic about showcasing its unfettered physicality.

Against Switzerland, its center back received a red card in the 80th minute for a dangerous tackle that denied a goal scoring opportunity. That followed a draw against Canada during which the goalkeeper went out to clear a ball with his fists and clocked a Canadian. While it wasn’t ruled a penalty because the keeper made contact with the ball first, it sparked widespread outrage because it appeared so violent.

“I know he wins the ball,” retired English legend Wayne Rooney said on BBC. “But the follow-through, he hits him in the temple.”

Bosnia has some history delivering knockout blows to giants. Despite being ranked No. 61 in the world, it reached this tournament by eliminating four-time champion Italy in the qualifying playoffs in March. On that night in Zenica, as red flares lit up the night sky, Bosnia stunned Europe by winning in a shootout.

And the penalty that clinched Bosnia’s spot in America came from none other than Esmir Bajraktarevic, Bosnia’s resident American. Bajraktarevic was born in Appleton, Wisc., made his professional debut in Major League Soccer, and the kid known as the “Milwaukee Messi” represented the U.S. at two different youth levels.

But in 2024, he put in a request to switch allegiance to his parents’ homeland. And now, he hopes to extend the Americans’ brutal run against teams from across the Atlantic.

The most recent U.S. victory over European opposition came against none other than Bosnia, way back in 2021. Since then, they’ve registered a pair of draws and then 10 consecutive defeats, falling to superpowers like Germany and lesser teams such as Slovenia.

Still, the Americans are strongly favored on Wednesday. Opta gives them a 78.5% chance of advancing, with a 42.5% likelihood of reaching the quarterfinals—which would potentially require beating another European team, Belgium, in the Round of 16.

But there’s only one number Pochettino and his players care about: 2.5%. That’s the chance the Americans are being given of pulling off the absolute shock of winning the World Cup.

Write to Andrew Beaton at andrew.beaton@wsj.com and Joshua Robinson at Joshua.Robinson@wsj.com