His sports career began when he was 7 years old, in a regional kart tournament. Between 2013 and 2014 he competed in championships and among other titles, he won the KF World Championship and became the youngest driver to obtain a title in the CIK-FIA World Championship.

In his first year in cars, Norris was crowned in a British F4 championship. He also won 2 of the 4 races he competed in at the BRDC Formula 4 Winter Trophy that year.

In 2016 he participated in Formula Renault 2.0, at the NEC and Eurocup championships with the Josef Kaufmann team, where he won both championships
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At the end of 2017 he competed in the 2018 F2 Championship where he won his debut race and took the podium eight other times.

After obtaining the European Formula 3 Championship, Norris entered the McLaren Young Drivers Program.

Since 2019, the British shares team with Carlos Sainz Jr.