Bettinelli was born in Camberwell, Greater London. He started his career with Fulham, signing his first professional contract in July 2010. Bettinelli joined Conference Premier club Dartford on 17 August 2012 on a one-month loan to gain first-team experience. He made his debut a day later, in a 2–0 away loss against Macclesfield Town. Having made seven appearances so far, his loan spell was extended for another month. This would happen again, as it was extended until Christmas. He went on to make 42 appearances in all competitions before being recalled by Fulham on 26 March 2013. Bettinelli joined League Two club Accrington Stanley on 28 August 2013 on a one-month loan. He made his debut the same day in the League Cup against Cardiff City in a 2–0 home defeat. His Football League debut on 31 August, in a 1–0 home defeat to Burton Albion. Fulham confirmed on 2 September 2013 that Bettinelli would stay with Accrington for the rest of the 2013–14 season, but he was recalled on 26 December and was named as a substitute in Fulham's 2–1 away win over Norwich City in the league as cover for David Stockdale. On 3 January 2014, he re-joined Accrington Stanley on loan until 30 January 2014.
2014–present
Bettinelli agreed a two-year extension to his Fulham contract on 10 January 2014, which would keep him at the club until June 2016. He made his debut for the club on 26 August in the second round of the League Cup away to West London rivals Brentford, keeping a clean sheet in a 1–0 victory. Four days later he made his league debut for Fulham in the Championship, a 1–1 home draw with Cardiff City. On 5 December, he was sent off after 18 minutes in a 5–0 loss to Watford at Craven Cottage for fouling Matěj Vydra, conceding a penalty that Troy Deeney put past substitute goalkeeper Gábor Király. Bettinelli signed another contract extension on 30 August 2017, keeping him at Fulham until June 2020 with an option in the club's favour to extend for a further year. He was second-choice to David Button for all but the last six matches of the 2016–17 Championship season and thought that his Fulham career was over, but manager Slaviša Jokanović eventually made him the starting goalkeeper for the following season. Fulham finished third and qualified for the play-offs, defeating Derby County 2–1 on aggregate in the semi-final and Aston Villa 1–0 in the final at Wembley Stadium. In October 2018, Bettinelli signed a new contract to last until 2021. For 2018–19, he lost his place to on-loan Spaniard Sergio Rico shortly before Jokanović's dismissal in November, and the situation remained the same under new manager Claudio Ranieri. He ended his season in February 2019 when he was ruled out for six months following a knee operation.
