Turgott joined West Ham United aged 8 becoming a youth and development squad player. He made his West Ham debut on 5 January 2014 in a 5–0 FA Cup defeat by Nottingham Forest. This was his only appearance for West Ham before being released in 2015.
Loan spells
He moved on loan from West Ham United to Bradford City in November 2012, and made his professional debut in the FA Cup in a 1–1 draw against Brentford on 30 November 2012. Turgott stated in December 2012 that he hoped the loan move would toughen him up. On 4 March 2013, he returned to West Ham after his loan spell ended, having made 11 appearances for the club. Turgott signed an initial one-month loan deal with Colchester United on 28 November 2013. Turgott made his Colchester United debut on 14 December 2013, coming on as a half time substitute in a 4–0 home defeat by Notts County. He scored his first club career goal on 26 December 2013, the opener in a 4–0 home win over Stevenage. Turgott returned to West Ham at the end of his loan having played four times and scoring once for Colchester. On 17 January 2014, Turgott joined Rotherham United on an initial one-month loan. He made his Rotherham debut a day later in a 3–0 away win against Shrewsbury Town coming on as an 88th-minute substitute for Ben Pringle. This was his only appearance for Rotherham before he was recalled by West Ham in February 2014. On 27 March 2014, Turgott joined League Two side Dagenham & Redbridge on loan for the remainder of the 2013–14 season. Turgott made his Dagenham & Redbridge two days after signing for the club, making his first start for them, in a 1–0 win over Oxford United. Turgott went on to make five appearance for the club.
Coventry City
Having been released from his contract by West Ham United, on 2 February 2015, Turgott signed for Coventry City on a short-term deal. Turgott made his Coventry City debut on 3 March 2015, coming on as a substitute for Adam Barton in the second half, in a 1–0 loss against Barnsley. Turgott scored his only Coventry goal on 28 March in a 0–1 away win against Peterborough United following-up a shot from Sanmi Odelusi. He was released by Coventry at the end of the 2014–15 season.
Turgott joined League Two club Leyton Orient on a one-year contract on 19 June 2015. He made his debut as a substitute on the opening day of the season, coming on for Jay Simpson in the 2–0 home win over Barnet, and scored his first goal for the club in the 3–0 win over Stevenage on 18 August. His opening goal for Leyton Orient was described as a "stunning goal" by their manager, Ian Hendon. Meeting a right-wing cross from Sean Clohessy, Turgott hit a first-time volley from past the Stevenage goalkeeper, Chris Day. In May 2016, he was released from Leyton Orient when it was announced that he would not be retained when his contract expired.
After attending the V9 Academy, on 27 June 2017, Turgott joined League Two side Stevenage. After a single appearance for the Boro, as a second-half substitute in an EFL Cup win at Millwall, he was loaned to Boreham Wood on 8 September 2017. He played the full 90 minutes against former club Leyton Orient the following day. On 21 December 2017, Stevenage announced that Turgott's contract had been terminated, following which he returned to Boreham Wood on a permanent deal.
In April 2017, Turgott was charged with offences relating to fraud and money laundering. He was subsequently cleared of a charge of converting criminal property, while a jury failed to reach a verdict on the other charge of fraud.
