By 10, Bacrot was winning junior competitions, and in 1996, at 13 years of age, he won against Vasily Smyslov.

He became a Grandmaster in March 1997 at the age of 14 years and 2 months, making him the youngest person at the time to have held the title until Ruslan Ponomariov took the record that December.

In 2023 Bacrot took part in the Chess World Cup in Baku as the 52nd seed, making it to the fourth round after defeating Nay Lin Tun, Aleksandar Indjic and the 13th seed Yu Yangyi from China, before losing to Indian GM Vidit Gujrathi.

He, then, participated in the FIDE Grand Swiss 2023, where he scored 6,5/11 and was in contention for a top 2 finish until round 8 when he lost against Fabiano Caruana after blundering in time pressure.

Bacrot has played several matches against prominent players in his home town of Albert.