The Diamondbacks selected Bradley out of high school seventh overall in the 2011 MLB Draft, and he began his first full professional season with the Single–A South Bend Silver Hawks.

After a 2015 debut in which he outpitched reigning Cy Young Award champion Clayton Kershaw, Bradley's first major league season was derailed by injuries: first a sinus fracture from being hit with a line drive, followed by shoulder tendinitis.

He was a hyperactive child, and his mother, the local school principal, would place him in classes with teachers that she believed could handle his excess energy.

His friend and Owasso pitching rival Dylan Bundy was selected by the Baltimore Orioles fourth overall, making 2011 the first year that two Oklahoma high school students were drafted in the first round since 1973.

[8] At the time, Bradley had already committed to play baseball and football at the University of Oklahoma, and waited until the August 16 deadline to officially sign a $5 million, five-year contract with the Diamondbacks.

[9] Bradley spent the remainder of the 2011 season with the Missoula PaddleHeads of the Pioneer League, at that time the Rookie Advanced minor-league affiliate of the Diamondbacks.

[10] Bradley started the 2012 season with the Single–A South Bend Silver Hawks, taking the loss in his professional debut on April 6 after giving up one run in five innings of work to the Bowling Green Hot Rods.

[13] After Trevor Cahill was traded to the Atlanta Braves, Diamondbacks chief baseball officer Tony La Russa announced that Bradley would be in the team's 2015 starting rotation.

He outpitched Clayton Kershaw, becoming only the fourth rookie starter since 2003 to win against a reigning Cy Young Award winner in his major league debut.

[23] On April 28, Bradley suffered a sinus fracture when he was hit in the face by a line drive off the bat of Colorado Rockies outfielder Carlos González, and he was placed on the 15-day disabled list the next day.

[27] In 2017, the Diamondbacks had an unusually strong starting rotation, including Zack Greinke, Patrick Corbin, and Robbie Ray, and, rather than sending Bradley back down to the minor leagues, the team moved him to the bullpen.

He chose to remain in the bullpen even after Shelby Miller was optioned to the minor leagues, and began serving as the Diamondbacks' setup man.

[27] Going into the 2018 season, Diamondbacks general manager Mike Hazen said that, while Bradley's role in the pitching lineup had not been formally addressed, it was likely that he would remain a reliever.

[35] His difficulties throughout the remainder of the season were later attributed to a cracked fingernail on his right index finger, which made it difficult for him to grip his signature curveball and forced him to rely more on his fastball.

[36] The Diamondbacks' bullpen collapsed in September, with Bradley, Jake Diekman, Matt Andriese, and Andrew Chafin all allowing at least as many runs as innings pitched.

[38] The following year, Bradley revealed to Arizona Sports 98.7 FM that, after giving up 32 hits, 16 walks, and 20 earned runs in May and June, Lovullo and Hazen called him in for a meeting and discussed how to return him to form without demoting him to the minors.

[44] He made his team debut on September 1, pitching 1+2⁄3 shutout innings in a 16–2 loss to the St. Louis Cardinals less than three hours after his plane arrived in Cincinnati.

[45] In the postseason, Bradley took the loss in the first game of the 2020 National League Wild Card Series, giving up two hits in the 13th inning against the Atlanta Braves.

[49] On March 31, however, manager Joe Girardi said that Neris would maintain the ninth-inning position, with Bradley and José Alvarado serving as setup men.

[54][55] On June 26, Bradley broke his elbow while climbing over the dugout railing during a brawl in a game against the Seattle Mariners, sidelining him for at least one month.

[64] Bradley is a power pitcher who was praised for a successful transition from the starting rotation to the bullpen, working in both the eighth and ninth inning since 2017.

[68] After being reassigned to the bullpen in 2017, Bradley began to grow a distinct beard, leading to jokes about the necessity of facial hair on a reliever.

He clarified that he believed most of the improvement was due to the shorter inning workload placed on a reliever, but that, "Just in case the palm reader was correct, I won't shave my beard until I retire".

[72] In 2017, Bradley purchased a 500 acres (200 ha) ranch in Pawnee, Oklahoma, about 8 miles (13 km) away from the Spring Valley Rod and Gun Club.

Bradley and his high school baseball teammate, Mak Monckton, connected with the gun club owner and started Crash Landing Outdoors, a hunting guide company.