Mészáros began competing in senior international competitions in 2019. He competed at the 2019 European and 2019 World Championships but did not qualify for any finals.[9][10] At the 2020 European Championships, he helped the Hungarian team win the bronze medal behind Ukraine and Turkey.[11] In the event finals, he finished sixth on floor exercise and pommel horse and fifth on parallel bars.[12]
Mészáros began the 2022 season at the Osijek World Challenge Cup, finishing eighth on floor exercise, fourth on pommel horse, and fifth on parallel bars.[14] Then at the Koper World Challenge Cup, he won the bronze medal on the pommel horse.[15] He helped the Hungarian team finished sixth at the 2022 European Championships, and he finished sixth in the all-around.[16][17] In the floor exercise final, he won the silver medal behind the defending Olympic champion Artem Dolgopyat.[18] This marked the first time a Hungarian gymnast won a European silver medal on floor exercise, after Róbert Gál and György Guczoghy won bronze medals on the event.[19] At the Szombathely World Challenge Cup, Mészáros won his first FIG World Cup title on the horizontal bar.[20] Then at the World Championships, he once again finished 14th in the all-around final.[21]
Mészáros finished 17th in the all-around final and fifth in the horizontal bar final at the 2023 European Championships.[22][23] He won four medals at the 2023 Tel Aviv World Challenge Cup- gold on floor exercise and vault and silver on pommel horse and parallel bars- making him the most decorated gymnast of the event.[24] Then at the Osijek World Challenge Cup, he won the bronze medal on floor exercise.[25] He then won three medals at the Szombathely World Challenge Cup- gold on floor exercise, silver on parallel bars, and bronze on horizontal bar.[26] At the end of the World Challenge Cup series, he was ranked first on parallel bars and top five on every event besides still rings.[27] Mészáros finished 18th all-around in the qualification round of the 2023 World Championships.[28] As one of the top eight gymnasts not from a country that qualified a full team, he earned an individual quota for the 2024 Olympic Games.[29] He finished 11th in the all-around final with a total score of 81.665.[30]
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