Bouchard experienced championship success early in his junior hockey career when he helped London win the 2016 Ontario Hockey League championship and Memorial Cup in his first full season with the Knights, who chose him with the No. 17 pick in the 2015 OHL Priority Selection. He was named captain for the 2017-18 season, which he ended as a finalist for the Max Kaminsky Trophy as Defenseman of the Year and the Red Tilson Trophy as MVP of the OHL.

The Oilers selected Bouchard with the No. 10 pick in the 2018 NHL Draft and signed him to a three-year, entry-level contract on July 17, 2018. He became the youngest defenseman in Oilers history to score a goal in a 4-1 win against the Washington Capitals on Oct. 25 after turning 19 five days before the game. He was returned to London on Nov. 2 after playing seven NHL games. He had 53 points (16 goals, 37 assists) and 21 (four goals, 17 assists) in 11 playoff games for the Knights and won the Max Kaminsky Trophy as the league's top defenseman. He was also an alternate captain for Canada at the 2019 IIHF World Junior Championship, where he had three assists in five games.

Bouchard secured his place with the Oilers in 2021-22 with 12 goals and 31 assists in 81 games and helping them reach the Western Conference Final. He began the 2023 Stanley Cup Playoffs by becoming the second defenseman in Oilers history with at least eight points in a series (Paul Coffey, who did so four times).

Bouchard's 20 points (five goals, 15 assists) in the first 12 games of the 2024 playoffs were the most by a defenseman through the first two rounds of a single postseason in NHL history. He finished with 32 to lead defensemen and helped the Oilers to Game 7 of the Stanley Cup Final, a 2-1 loss to the Florida Panthers.

The following season, Bouchard had 67 points (14 goals, 53 assists) in 82 regular-season games and 23 points (seven goals, 16 assists) in 22 Stanley Cup Playoff games, and signed a four-year, $42 million contract with the Oilers on June 30, 2025, in lieu of becoming a restricted free agent. He ended the 2024-25 season tied for second in NHL history in points per game among defensemen in the playoffs (1.08, minimum 15 games), behind only Bobby Orr (1.24).

Bouchard scored his first NHL hat trick and had three assists in his 400th NHL game, Edmonton's 6-5 overtime win against the Washington Capitals at Rogers Place on Jan. 24. 2026, which included the secondary assist on Zach Hyman's goal at 19:28 of the third period and the primary assist on Connor McDavid's goal 47 seconds into OT. The hat trick was the fourth by an Oilers defenseman and first since Marc-Andre Bergeron on Jan. 14, 2006.