Bozek was drafted by the Los Angeles Kings in the 3rd round, 52nd overall in the 1980 NHL Entry Draft while playing for Northern Michigan University. Bozek was named a CCHA First-Team All-Star twice in his three years at Northern Michigan, leading the CCHA in scoring during the 1980–1981 season. In 1981, he was named a First-Team All-American - the first hockey All-American ever for Northern Michigan - after dominating college hockey with 90 points in just 44 games, and was a finalist for the Hobey Baker Award.
Bozek turned pro after his junior year, breaking straight into the Kings lineup for the 1981–82 season. As a rookie, he received an opportunity to spend a large portion of the season on the famed Triple Crown Line with Marcel Dionne and Dave Taylor due to an injury to Charlie Simmer, and he set a Kings rookie record (since broken) with 33 goals. In the playoffs, he secured a place in Kings lore by scoring the tying goal with 5 seconds left in the famed Miracle on Manchester game against the Edmonton Oilers, which was the greatest playoff comeback in NHL history as Los Angeles rallied from a 5–0 deficit to win 6–5 in overtime.
1982–83 would be a difficult year for Bozek, as he struggled with injuries and the sophomore jinx, finishing with 13 goals and 26 points in 53 games. At the conclusion of the season, he was dealt to the Calgary Flames. Always a primarily offensive player to that point in his career, in Calgary he began to develop the defensive side of his game and became a strong checker and penalty killer. However, his five seasons in Calgary would be blighted by nagging knee problems, and he was only able to play more than 70 games in a season once. His best year as a Flame was 1985–86, as he scored 21 goals and 22 assists for 43 points in 64 games, including a club-leading 4 shorthanded goals. He added 8 points in 14 games in the playoffs as Calgary reached the Stanley Cup finals.
After missing most of the 1987–88 season with a knee injury, Bozek was traded to the St. Louis Blues at the trade deadline along with Brett Hull in a deal for Rob Ramage and Rick Wamsley. Another knee injury limited him to 7 regular season games with the Blues, and he finished the season with just 3 goals and 10 points in 33 games between Calgary and St. Louis.
Bozek's time in St. Louis would be brief, as he was dealt to the Vancouver Canucks in a three-way deal prior to the 1988–89 campaign. He would have a strong bounce-back season in Vancouver, finishing with 17 goals and 35 points, and his defensive play was a factor in Vancouver improving to 3rd in the league defensively from 17th the previous season. He spent two more seasons in Vancouver, recording solid totals of 14 and 15 goals, although as usual he was slowed down by injuries which caused him to miss 40 games over that stretch. Due to his enthusiastic and scrppy style of play, he earned the nickname Steve "Boom-Boom" Bozek, a nickname coined by CKNW commentator Tom Larschied. He was selected to play for Canada at the 1991 World Championships, winning a silver medal.
In 1991, Bozek signed as a free agent with the expansion San Jose Sharks, and was a member of their inaugural team during the 1991–92 season, finishing with 8 goals and 16 points in 58 games. He then played for a season in Italy before retiring in 1993.
Bozek finished his career with totals of 164 goals and 167 assists for 331 points in 641 games, along with 309 penalty minutes. He never played a game in the minor leagues