Born in Eastleigh, Hampshire, Targett joined Southampton's academy at the age of eight, after being scouted by his local team.[4] He was named in a senior matchday squad for the first time on 24 September 2013, and was unused as they defeated Bristol City 2–0 in the third round of the League Cup.[5] On 26 April 2014, he was named in a Premier League game for the first time, remaining on the bench as Southampton won 2–0 at home against Everton.[6]