Born in Boca Raton, Florida, Brabham started racing karts in 2001 at the age of 7, competing in the Australian 'Midget' class. In 2007 he won the Queensland State 'Rookie' Title, following up by winning both the New South Wales and Queensland State Junior National Light Titles and finishing second in the Australian and Melbourne City Titles in 2008. In 2009, Brabham won the 'Junior Rotax' Young Guns Title, before graduating to Formula Ford racing towards the end of the year. In 2010 he competed in the Australian Formula Ford Championship with the CAMS Rising Star Team. For 2011, Brabham competed in the Victorian State Formula Ford Series and the Australian National Championship with Sonic Motor Racing Services. In the state series he won 8 out of 12, including 7 in a row and in the national series he scored 2 wins, 2 seconds and a pole position.
For 2012, Brabham joined Cape Motorsports with Wayne Taylor Racing, to compete in the U.S. F2000 National Championship. Brabham won the title over his teammate Spencer Pigot by seven points, capturing four wins from 11 podium finishes in the fourteen race season. In doing so he wins a $350,000 USD scholarship through the Road to Indy to race in the Star Mazda Championship in 2013. He signed to drive the no. 27 car for Andretti Autosport in 2013. Brabham won the series championship with two races remaining capturing a series-record 11th win in his rookie season. Brabham's championship secured a scholarship to compete in Indy Lights in 2014. On October 19, Brabham and Andretti Autosport announced that they would remain together to challenge for the Indy Lights title in 2014. He won one race and collected four podiums and 10 top 5s in 14 races to finish fourth in the overall standings.
In November 2014 Brabham competed in the second round of the inaugural Formula E world championship in Putrajaya, Malaysia. He replaced Charles Pic for Andretti Autosport. At age 20 Brabham became the youngest driver to compete in Formula E.
In 2015, Brabham made his Speed Energy Formula Off-Road debut at Honda Indy Toronto. He also competed at the Castrol Gold Coast 600, Valvoline Raceway (Sydney Speedway) and the Sydney 500 V8 Supercars event in the Stadium Super Trucks. He returned to the series in 2016 at the Adelaide Street Circuit securing a podium, which he backed up with a podium at St Petersburg in support of the IndyCar Series. In June 2016, Brabham swept the Detroit Belle Isle Grand Prix rounds. He later joined Team Traxxas, driving the No. 83 alongside Sheldon Creed. He finished second in the 2016 standings, 75 points behind teammate Creed.
In 2016, Brabham made his Indianapolis 500 debut, driving the No. 61 for PIRTEK Team Murray. He qualified 26th and finished 22nd.
In 2017, Brabham returned to Australia to race in the Supercars Championship, making his debut at the 2017 Perth SuperSprint for Lucas Dumbrell Motorsport in place of Taz Douglas.
(key) (Races in bold indicate pole position; races in italics indicate fastest lap)
(key) (Bold – Pole position. * – Most laps led.)
* Season still in progress.