[6] Thorton was selected by the Houston Astros in the fifth round of the 2015 Major League Baseball draft, and he received a $325,000 bonus when he signed.

[7][8] He made his professional debut that summer with the Low-A Tri-City Valley Cats, going 4–0 with a 3.27 ERA in 12 starts and 3 relief appearances.

[5] On November 17, 2018, Houston traded Thornton to the Toronto Blue Jays for utility player Aledmys Díaz.

[12] After 2019 spring training, the Blue Jays announced that Thornton had made the Opening Day roster on March 26.

[14] Thornton earned his first career win on May 14 on the road against the San Francisco Giants, allowing two runs and striking out seven in 52⁄3 innings.

[15] Thornton's rookie season ended with a 6–9 record and 4.84 ERA in 29 starts, the most in his big league career, and 3 bullpen outings.

[16] He lowered his ERA below 5.00 in the final month of the season,[17] crediting his development in part to rotation-mates Clay Buccholz and Clayton Richard.

[27] On January 13, 2023, Thornton signed a one-year, $1 million contract with the Blue Jays, avoiding salary arbitration.

[31] The Mariners added Thornton to their roster on August 1, and he became a bullpen regular, pitching in 23 of the team's final 56 games.

[8] Thornton picked up his first MLB save on June 26, 2024, relieving closer Andrés Muñoz with no outs and the bases loaded, allowing just one inherited runner to score.

[35] Thornton added one mile per hour to his fastball velocity, compared to 2023, and he was in the top 12 percent of pitchers at getting batters to swing at pitches thrown outside the strike zone.