Baz won the European Superstock 600 Championship in 2008 on a Yamaha YZF-R6, making an immediate mark at European level as a teenager. He followed this with four consecutive seasons in the FIM Superstock 1000 Cup from 2009 to 2012, finishing eighth in both 2009 and 2010 on Yamaha machinery before switching to Honda for 2011 and Kawasaki for 2012. He also competed in the British Superbike Championship during this period.
Baz entered the Superbike World Championship, and Kawasaki Racing Team confirmed him for the 2013 season. His WorldSBK debut period placed him among the developing talents in the series before his move into MotoGP.
On 4 October 2014, it was announced that Baz would move into MotoGP with Forward Racing for the 2015 season. His best standalone result came at Misano in September, where changeable conditions helped him to fourth place. He finished his debut MotoGP season seventeenth in the standings with 28 points.
Baz rode for the Avintia Racing team in 2016 and 2017, competing consistently in the satellite Ducati ranks. His most notable return to MotoGP action came at the 2018 British Grand Prix, where he replaced the injured Pol Espargaró at the factory KTM team. The race was ultimately cancelled due to dangerous track conditions following heavy rain and resurfacing work, denying him what would have been a high-profile single outing.
For the 2018 season, Baz returned to WorldSBK with the Gulf Althea BMW Racing team. His World Superbike career continued in subsequent seasons, including a brief role as a replacement rider — filling in for the injured Chaz Davies at Jerez in September, finishing sixth in race one and ninth in race two.
Baz moved to the United States for competitive opportunities in MotoAmerica. In 2021 he joined the HSBK Racing Ducati team, riding a Ducati Panigale V4 R, and finished the season fourth in the championship standings. By 2024 he was competing for the War Horse Ducati team in the Steel Commander MotoAmerica Superbike Championship.
Alongside his MotoAmerica commitments, Baz entered the King of the Baggers series in 2025 with the Indian Motorcycle S&S Cycle factory team, a touring-derived class that has grown in popularity within the American racing scene.
Baz also competed in FIM Endurance World Championship events. At the final round of the 2021 season at Most in the Czech Republic, he rode for the Moto Ain team on a Yamaha YZF-R1 alongside Randy De Puniet and Corentin Perolari. The team finished twenty-first after mid-race damage delayed their progress.
Baz's career path — European Superstock champion, WorldSBK, MotoGP, back to WorldSBK, then trans-Atlantic competition in MotoAmerica and King of the Baggers — reflects the varied routes available to European-trained riders seeking competitive opportunities across different eras and regulations. His 2008 Superstock title as a fifteen-year-old remains the most clear-cut championship achievement of a career defined by versatility across multiple series and manufacturers.