Burch started his career at Tottenham Hotspur, where he rose through the ranks of the youth academy before making his first-team debut in 2003. However, he struggled to secure a regular spot in the starting XI due to stiff competition from other goalkeepers at the club. As a result, Burch went on loan to several lower league clubs, including Woking, Barnet, and Sheffield Wednesday, to gain more playing time and experience.