Cassidy began karting at the age of six in New Zealand and remained in karts until 2010. He competed in midget car racing from the age of eight, developing racecraft outside the traditional European ladder system. He entered formula car racing through New Zealand and Australian Formula Ford championships; in 2009 he was runner-up in the New Zealand Formula First Championship and in 2010 runner-up in the New Zealand Formula Ford Championship, earning Rookie of the Year honours on both occasions.
In 2011, Cassidy raced in the Toyota Racing Series with Giles Motorsport, taking two race wins on the final weekend and finishing runner-up in the championship to his teammate Mitch Evans. He subsequently competed in the ADAC Formel Masters, Formula Abarth, and the Fujitsu V8 Supercar Series before establishing his career base in Japan.
Cassidy spent the mid-2010s building an exceptional record in Japanese categories. He competed in both Super Formula and Super GT simultaneously, and in 2017 he won the Super GT championship. In 2018, he narrowly lost the final-round title battles in both Super Formula and Super GT in consecutive weeks, finishing second in each series. He returned to claim the Super Formula title in 2019, completing what he described as the triple crown of Japanese motorsport. That victory established him as one of the most successful foreign drivers in Japanese racing history.
Cassidy made his DTM debut in the final round of the 2021 season at the Norisring, stepping in for Red Bull Racing reserve driver Alex Albon in a combined entry run by AF Corse and Red Bull. He drove Red Bull's Ferrari 488 GT3 for that appearance. He returned for the full 2022 DTM season with the same Ferrari-based programme, gaining experience in the German touring car series alongside his Formula E commitments.
Cassidy made his Formula E debut in the 2020–21 season with Envision Virgin Racing, replacing Sam Bird and partnering Robin Frijns. After a slow start he scored his first pole position at a wet-weather session in Rome, though he subsequently spun from the lead due to a software fault. He achieved his maiden Formula E podium at the Puebla ePrix and added a further podium in New York City, finishing 15th in the championship.
In 2021–22, Cassidy converted his New York City pole position into his first Formula E victory, benefiting from sudden rain that caused the race to be aborted and classified by countback. He finished 11th in the championship. The 2022–23 season proved his most competitive, with Cassidy winning in Berlin, Monaco, and Portland and entering the final London rounds as a championship contender. A teammate collision caused by miscommunication dropped him from the lead and effectively handed the title to Jake Dennis. Cassidy finished the season second in the championship, 20 points off the title.
For 2023–24, Cassidy joined Jaguar TCS Racing on a multi-year deal, partnering Mitch Evans. He scored two victories and six additional podiums in the first half of the season, building a 25-point lead heading to the Portland ePrix. He spun from the lead on the penultimate lap at Portland and left the event without points; further misfortune in London condemned him to third in the final standings. He departed Jaguar following the 2024–25 season, again finishing as championship runner-up. In September 2025, he was confirmed at new team Citroën Racing for the 2025–26 campaign alongside Jean-Éric Vergne.
Cassidy has balanced his circuit-racing programme with endurance commitments. He was named to drive for Peugeot Sport in the Hypercar class of the FIA World Endurance Championship for 2026.
Cassidy's record in Formula E — two championship runner-up finishes and multiple race victories with three different teams — marks him as one of the series' most consistently fast drivers across the Gen2 and Gen3 eras. His Japanese titles in both Super GT and Super Formula, achieved in the same period, underline his adaptability across radically different car types. He is one of a small number of drivers to have competed seriously and won in Japanese domestic championships, DTM, Formula E, and WEC-class endurance racing within a single career.