Branch was born in Johannesburg but moved to Jwaneng in Botswana three days after birth when his father Kevin, a Debswana employee, took up a posting there. He grew up in Botswana and attended Krugersdorp High School in Gauteng. As a child he attended the final stage of the 1996 Paris to Cape Town Dakar Rally with his father, an experience he credits with planting the ambition to one day compete in the race himself.
He began riding motorcycles at the age of 10 and made his first race start at the age of 5. At 18 he was awarded South African colours for motocross.
Between 2003 and 2005, Branch competed overseas in Europe and the United States. He won the Southern Pokal Championship in Germany in 2003, finished second in the Russian Race Series in 2003, won the Torgau 24 Hour Endurance Race in Germany in 2005, and finished third at the Denver Supercross in 2005. During this period he also scored world championship points in four motocross Grands Prix.
Branch won the African Motorcycle Union continental motocross championship five times — in 2003, 2007, 2009, 2011 and 2016. He was South African OR1 Cross Country Champion in 2016, 2017 and 2018, and Overall Cross Country Champion across those same three consecutive seasons. He won the Botswana 1000 Desert Race seven times, establishing a reputation as a desert racing specialist. He also won the Khawa Dune Challenge four times — in 2013, 2014, 2015 and 2017 — and was a Gold finisher at the Roof of Africa endurance event in 2011.
His mother Glenda, who had been a major influence on his racing career, died from cancer in 2013. Branch has spoken publicly about racing in her memory following her death.
Branch made his Dakar debut in 2019, riding for KTM, and finished 13th overall — winning the Rookie category. In 2020, riding for Hero MotorSports, he won Stage 2 to become the first Botswanan to win a Dakar stage and finished 21st overall. His two appearances with Yamaha, in 2021 and 2022, both ended in retirement. Returning to Hero for 2023, he finished 26th and won two stages. In 2024 he produced his finest Dakar result, finishing second overall — just ten minutes behind winner Ricky Brabec — and winning Stages 1 and 11. The 2025 Dakar ended in retirement following a crash and subsequent mechanical problems.
Branch won the FIM World Rally-Raid Championship title in 2024. The 2025 season was disrupted by a concussion suffered during that year's Dakar, mechanical failures at subsequent rounds, and a crash in Portugal that prevented him from challenging for the championship title. As of the 2026 season he continued to compete in the Rally GP class with Hero MotorSports.
Branch works part-time as a commercial airline pilot in Botswana, based in Gaborone.