Vesti started his career at the age of fourteen in Danish Formula Ford, in which he managed to finish fourth in the championship.

[24] At the end of the season, Vesti again raced in the Macau Grand Prix, but this time was called up to replace an injured Jehan Daruvala at Prema Powerteam.

[45] He made a brilliant charge in Race 1, overtaking Théo Pourchaire on the third last lap and claiming his second win overall whilst helping Prema become the teams' champions.

Despite climbing to third by lap 9, his effort would be undone as he touched teammate Sargeant, damaging Vesti's front wing and taking both out of the race.

[53] Vesti would sign for ART Grand Prix to race in Formula 3 alongside Russian Aleksandr Smolyar and American Juan Manuel Correa.

[69][70] He thus overhauled Victor Martins to repeat his previous year's championship finish of fourth place, claiming one pole and a win each, and four more podiums.

[71] Vesti got his first taste of Formula 2 machinery when he participated in the post-season test in 2021 with ART Grand Prix alongside former F3 title rival Théo Pourchaire.

[75][76] During the third round in Imola, in the feature race, Vesti started on the soft tyres, which allowed him to pit during an early safety car.

With many rivals ahead on the alternate strategy, Vesti was able to claim his first points of the year by finishing sixth, after which he stated that he needed to "work on qualifying".

Late in the race, he seized on a mistake from Jehan Daruvala during the safety car restart to take the lead in the closing laps and claim his maiden Formula 2 victory.

[82][83] Vesti stalled at the start of the feature race, but due to many rivals being involved in incidents, he stayed clean to secure seventh place.

[87] Vesti scored his first pole position of the season at the Red Bull Ring, vanquishing his previous qualifying struggles.

[90] Vesti fitted wet tyres on a drying track and the strategy failed to work, which resulted in him falling down to 14th by the end of the race.

[93] In the feature race, he lost a position to Jack Doohan at the start but regained third place after the Australian spun whilst fighting teammate Pourchaire, adding another podium.

[100] In the feature race, a timely pit stop saw Vesti in a net position fourth place following the lap 11 safety car restart.

He would gain positions when Marcus Armstrong had to serve a penalty, and Vesti also passed Ayumu Iwasa, earning him another second place as he would run out of time to catch winner Jehan Daruvala.

Vesti ended up ninth in the championship with 117 points, helping ART to third in the teams' standings whilst scoring one win, one pole, and five podiums.

[104] Vesti qualified fifth for the Bahrain opening round, but was penalised three places for the sprint due to impeding Arthur Leclerc.

In the sprint race, he remained out of trouble and even lead after a late safety car, but made a mistake at a corner, dropping to second whilst teammate Bearman won.

[119] Vesti qualified in eighth for Barcelona and from third, stormed past Jak Crawford and Amaury Cordeel to lead the sprint race and ultimately take back-to-back wins from there.

He would hold the net race lead by a commanding distance, but a late safety car dented his hopes, as Verschoor and Iwasa on the alternate strategy pitted for fresher tyres.

[125][126] Having extended his lead to 20 points, he qualified in tenth in Silverstone, and drove away to his fourth victory of the year in the sprint race by 13 seconds.

Starting on the reverse front row for the sprint race, he challenged Enzo Fittipaldi into turn 1 but fell back to fourth after being overtaken by Isack Hadjar and Verschoor.

He fought back and passed Hadjar and Verschoor before overtaking Fittipaldi with three laps to go to take his sixth win on the season.

He charged back up the order, and despite a late last lap collision with Zane Maloney that saw his car launched into the air and nearly into the barriers, he finished in third place.

[145] Despite scoring the most wins out of all the drivers in 2023 he missed out on the title by eleven points as Pourchaire finished in fifth place but was still praised for his great driving skills throughout the season.

Vesti also competed in his first 24 Hours of Le Mans that year with Cool Racing alongside Matt Bell and Naveen Rao.

[152] For 2025, Vesti signed with Whelen Cadillac Racing to compete in the 2025 IMSA SportsCar Championship, alongside Earl Bamber and Jack Aitken.

[164] He drove the Mercedes W15 for the first time that year during the Yas Marina young drivers' test during the morning session, where he finished in 21st place overall.

[165] In May 2024, Vesti sampled a Formula E car for the first time at the Tempelhof Airport Street Circuit during the Berlin rookie test, where he drives for Mahindra Racing.