[4] Palmer competed in the BRDC Formula 4 Winter Championship in 2013, ahead of graduating to the category for 2014.

A promising debut full season in BRDC F4 saw Palmer take two wins and finish sixth in the 2014 overall standings.

[4] After his record-breaking BRDC F4 season, Palmer was nominated for the 2015 McLaren Autosport BRDC Award alongside Jack Aitken, Ben Barnicoat, Jake Hughes, Ricky Collard and Toby Sowery.

After two days of assessments at Silverstone, Palmer was named the winner at the Autosport Awards in London, receiving the prize of £100,000, a McLaren F1 test, Arai Carbon helmet and BRDC membership.

[8] He finished as runner-up to Sacha Fenestraz with wins at Silverstone, Monza and Monaco and another seven podiums.