Following nine years in the youth set up at Charlton Athletic, and a season with Fisher Athletic, 16-year-old Oztumer joined Manisaspor in 2008, who at the time were in the Turkish Süper Lig. Despite featuring mainly for the Manisaspor youth team, in 2009 he earned a move to Sivasspor, also of the Turkish Süper Lig, where he signed his first professional contract. Oztumer never played for the first team at Sivasspor, but played 29 times for the reserve team in A2 Ligi, scoring eight goals. In 2011, Oztumer left Sivasspor to join Anadolu Üsküdar in the TFF Third League, the fourth tier of Turkish football, where he played senior footballfor the first time, playing 34 league games in 16 months, scoring five goals. Before the start of the 2012–13 season, Oztumer returned to England, joining Isthmian League side Dulwich Hamlet. He went on to score over 60 goals from midfield in all competitions over two seasons, during which time Dulwich won promotion to the Isthmian League Premier Division. In June 2014, he moved back into the professional game, joining Peterborough United for an undisclosed fee, leaving with the best wishes of Dulwich Hamlet manager Gavin Rose. He played one friendly in pre-season, scoring a free kick against St Neots Town, before suffering an injury which kept him out of the early part of the 2014–15 season. He made his Peterborough debut as a substitute in a 2–2 draw with Oldham Athletic on 4 October 2014. He made his first start for the club on 13 December 2014, scoring the winning goal away to Leyton Orient. On 17 June 2016, Oztumer signed for Football League One side Walsall on a two-year contract upon the expiry of his contract at Peterborough. He scored on his debut in a 3–1 win over AFC Wimbledon on 6 August 2016. At the end of the 2017–18 season he announced his decision to leave Walsall due to his contract expiring. On 13 June 2018, Oztumer signed for EFL Championship team Bolton Wanderers on a two-year deal. He scored on his debut, in a 1–2 defeat to Leeds United in the EFL Cup on 14 August. On 15 August 2019 he left the club after handing in his notice due to the club's ongoing financial problems. On 16 August 2019, Oztumer was announced as a free signing for Charlton Athletic on a two-year deal.
