At age 17, he reached the final of the 2013 EBSA European Under-21 Snooker Championship, losing 0–6 to James Cahill.

[1] He achieved his first significant professional success when he defeated Ken Doherty and Neil Robertson in televised matches at the 2020 Championship League, topping his group.

[2][4] At the 2020 World Snooker Championship, he defeated Ross Muir, Jimmy Robertson, and Robert Milkins in the qualifiers to reach the main stage of the event for the first time, but his Crucible debut ended with a 7–10 first-round defeat to Stuart Bingham.

[6] Carty finished the 2021–22 season ranked 75th in the world, and was relegated from the professional tour.

[10] At the 2023 European Masters, he defeated Thepchaiya Un-Nooh 5–1 in the last 32 and Ricky Walden 5–3 in the last 16 to reach the first ranking quarter-final of his career, where he lost 1–5 to Judd Trump.