Flörsch was born in Grünwald, Bavaria, and grew up near Munich, attending Oberhaching Grammar School before settling in Pullach. She has cited Lewis Hamilton, Sebastian Vettel and Michael Schumacher as her racing idols. She began karting in 2005 and, from 2008 to 2014, competed in events across Europe including the SAKC Championship, the ADAC German Championship and the European Championship Easykart, becoming the first female and youngest competitor in each series. She was also scouted by Red Bull during her karting years. She has been a vocal critic of gender-segregated racing series, describing the W Series as "a step back on a sporting level" and labelling F1 Academy as "segregation."
In 2015 Flörsch raced in the Ginetta Junior Championship for HHC Motorsport. At Thruxton she became both the youngest driver to win a Ginetta Junior race and the first rookie to win two races in a single weekend. Financial issues cut her season short at the mid-season point; she was running third in the championship at that stage.
From 2016 Flörsch competed in ADAC Formula 4 with Motopark. In her debut race she became the first female to score points in an ADAC F4 event. In 2017 she drove a pink BWT-sponsored car for ADAC Berlin-Brandenburg, earning her first single-seater podiums at the Sachsenring and Hockenheim along with two fastest laps.
Flörsch's career was interrupted dramatically at the 2018 Macau Grand Prix. During the main race on lap four, contact with Jehan Daruvala caused a front-left suspension failure; the car launched off the car of Sho Tsuboi, broke through catch fencing and struck a photographers' bunker at high speed before coming to rest on its wheels. Flörsch was conscious but was diagnosed with a spinal fracture, requiring surgery lasting almost ten hours. The operation was reported as successful with no fear of paralysis. Two photographers, Tsuboi and a marshal were also hospitalised.
Flörsch had joined Van Amersfoort Racing mid-2018 in the FIA Formula 3 European Championship, scoring a single point at the Red Bull Ring before the Macau incident ended her season. After her recovery, she competed in the 2019 Formula Regional European Championship with Van Amersfoort Racing, scoring points in all 24 races and setting a fastest lap at the Red Bull Ring. She finished seventh in the standings.
In the 2020 FIA Formula 3 Championship with Campos Racing she was the first woman to compete in the series since its formation from the merged GP3 and European F3 categories, finishing 29th of 35 drivers with a best finish of twelfth. Returning for 2023 with PHM Racing by Charouz and as a new member of the Alpine Academy's Rac(H)er programme, Flörsch became the first female driver to score points in the FIA Formula 3 Championship when she finished seventh in a wet Spa-Francorchamps feature race; she had earlier had a points finish at the Red Bull Ring disqualified post-race. She remained with Van Amersfoort Racing for 2024 but failed to score points; teammate Noel León made the podium four times that season.
Flörsch made her prototype debut in 2020 in the European Le Mans Series LMP2 class with Signatech's Richard Mille Racing Team, an all-female crew. She contested the 2020 24 Hours of Le Mans with the same team alongside Beitske Visser and Tatiana Calderón, finishing ninth.
In 2021 she remained with Richard Mille for the FIA World Endurance Championship, helping the team to five points finishes and qualifying a career-best sixth at the 8 Hours of Bahrain. A late-season call-up to Algarve Pro Racing in the ELMS finale at Portimão resulted in a third-place finish alongside Ferdinand Habsburg and Richard Bradley — the first overall ELMS podium for a female driver since Natacha Gachnang in 2013.
For 2022 Flörsch joined G-Drive Racing for the ELMS in LMP2, but the team withdrew following the Russian invasion of Ukraine. Algarve Pro Racing took over the programme, and Flörsch and Bent Viscaal took a second place on debut at Paul Ricard. The remainder of the season brought limited results, and at the 24 Hours of Le Mans the team lost five laps to an early sensor failure before recovering to fifth in the LMP2 Pro-Am class. Flörsch returned to Algarve Pro Racing in a test and development role in 2025.
Alongside her 2021 WEC programme, Flörsch raced in the Deutsche Tourenwagen Masters for Abt Sportsline, becoming the only Audi to use Schaeffler's new Space Drive steer-by-wire system. A difficult first half gave way to eight points from the final six races. In December 2024 she signed with HMD Motorsports for the 2025 Indy NXT series, but budget difficulties forced her to withdraw after a single race. In 2026 she joined the Opel Formula E works team as test and development driver.
Flörsch won the World Comeback of the Year award at the 2020 Laureus World Sports Awards, in recognition of her return to racing following the spinal injury sustained at Macau in 2018.