[4] Jessup commenced his speedway career in 1969, when he rode for Eastbourne Eagles recording 6.62 average during the 1969 British League Division Two season.

The following season, he won the British League Division Two Riders Championship, held at Hackney Wick Stadium on 25 September 1970.

He finished runner-up in the Speedway World Championship to fellow countryman Michael Lee.

[11] He continued to perform at the highest level in 1981 and moved to join Wimbledon Dons for the 1982 British League season, where he topped the London team's averages for both 1982 and 1983.

He returned to King's Lynn for the 1984 British League season but struggled to retain the form he had for the previous decade.