Curry made his professional debut against Wasps on 4 November 2016 in the Anglo-Welsh Cup.[2] At the end of the 2016-17 season, he was named a joint recipient of Sale Sharks' Young Player of the Season award, sharing the award with his brother Tom.[3] In September 2020 Curry started for the Sale side that defeated Harlequins in the final of the Premiership Rugby Cup.[4]
In December 2024, he scored a try and won man of the match as Sale beat Bristol Bears 0–38 away at Ashton Gate, the first time Bristol had been kept scoreless in a Premiership fixture since September 2016.[5]
Curry was called up to the senior England squad by Eddie Jones for their 2017 summer tour of Argentina,[10] but did not make an appearance on the tour. In June 2021 he was included in the senior squad again[11] and on 4 July 2021 made his debut off the bench against the United States at Twickenham.[12]