After unsuccessful trials at Aston Villa and Cardiff City, Bowen started his career as a scholar at Hereford United. At the age of 16, Bowen broke into the Hereford United first team after impressing for the youth team that season. He scored his first career goal on 21 April 2014 in Hereford's 3–2 home victory over Alfreton Town.

After Hereford were expelled from the Football Conference in June 2014, Bowen signed for Premier League club Hull City in July. After impressing during pre-season in 2016, he made his first-team debut for Hull on 23 August, playing the full 90 minutes in a 3–1 EFL Cup victory over Exeter City. On 7 November 2016, Bowen signed a new two-year contract Hull.

Bowen scored his first goal for Hull on 5 August 2017, away to Aston Villa in a 1–1 draw. In September 2017, Bowen signed a new contract that tied him to the club until June 2020.
In his first full season in 2017-18 he scored 15 goals in all competitions which saw him finish as Hull's top scorer that season. At the annual end of season awards ceremony on 8 May 2018, Bowen won both the Hull City Supporter's Player of the Year and Players' Player of the Year awards. These wards were relatively insignificant has Bowen also picked up the seasons top prize the Hull City Official Supporters Club Player of the Year award.
Bowen was nominated for the EFL Championship Player of the Month award for December 2018, subsequently winning the award in January 2019.

In March 2019 he was selected to the 2018–19 Championship Team of the Season.
He scored 22 goals in all competitions during the 2018–19 season, which saw him finish as Hull's top scorer. This was the best return from a Hull City player since Stuart Elliot netted 29 times back in the 2004/05 season. At the annual end of season awards ceremony on 7 May 2019, Bowen once again won most of the awards picking up the Player of the Year, Hull City Supporter's Player of the Year and Players' Player of the Year awards. Once again the best and most prestigious award was to follow a little later when HCOSC not only presented him with the HCOSC Player of the Season award but also a cheque for Jarrod’s chosen charity – Life for a Kid.