Lindstrøm played youth football at several smaller clubs in the Vestegnen area of the Capital Region of Denmark, before joining Brøndby's academy where he made his first-team debut in 2018.

On 11 July, Lindstrøm made his first European appearance for Brøndby, coming on as a substitute in the first leg of their UEFA Europa League qualification round against Inter Turku.

[9] On 1 August, he came on as a substitute in the last minute of regulation and scored two goals in extra time as Brøndby defeated Lechia Gdańsk 4-1 (5-3 aggregate) in the second round of Europa League qualification.

[10] Due to his performances in the first months of the 2019–20 season, he received a lot of attention and was pointed out as the club's jóia da coroa (crown jewel) in a scouting report by Braga, Brøndby's opponent in Europa League qualifiers.

[11][12] After the COVID-19 pandemic had suspended the Superliga for some months, Lindstrøm scored the club's first goal in competition after the return, securing a 1–0 win over SønderjyskE on 2 June 2020.

[25][26] On 11 July 2021, it was announced via official channels that Lindstrøm had signed with Bundesliga club Eintracht Frankfurt on a five-year contract.

[31] After mixed performances to start the season, he recorded a last-minute assist to Jens Petter Hauge in an important 2–1 group stage victory in the UEFA Europa League against Olympiacos on 2 November.

[32] On 21 November, Lindstrøm scored his first goal for Eintracht Frankfurt in a 2–0 win over SC Freiburg, toe-poking the ball past Mark Flekken inside the area.

[33] He continued his strong performances in December, scoring in league appearances against Bayer Leverkusen, Borussia Mönchengladbach and Mainz 05 to earn him the award of Bundesliga Rookie of the Month.

[34][35] In the second half of the 2021–22 season, he grew into an undisputed starter in attacking midfield next to Daichi Kamada and behind lone striker Rafael Borré.

[36] In the Europa League, Lindstrøm reached the final with Frankfurt after knocking out Real Betis, Barcelona and West Ham United.

In the final on 18 May 2022 against Rangers, he featured in the starting eleven before being substituted for Hauge in the 70th minute, as Frankfurt won the title after a penalty shoot-out.

[50] On 29 September 2020, Lindstrøm was named in manager Albert Capella's squad for the 2021 UEFA European Under-21 Championship qualifiers against Malta and Finland.

[55] On 22 November 2022, he made his debut in a major tournament in Denmark's opening World Cup game against Tunisia, coming on as a replacement for Andreas Skov Olsen in the 65th minute of a 0–0 draw at Education City Stadium.