Cortese was born in Ochsenhausen in what was then West Germany. His father is Italian and his mother German. He began racing on pocket bikes at age nine and became both European Pocket Bike champion and German Mini Bike champion before progressing to larger machinery.
Cortese made his 125cc World Championship debut in 2005 with Kiefer-Bos-Castrol Honda, scoring 8 points in his first season. He moved to Elit – Caffè Latte Honda in 2006, teaming with reigning 125cc world champion Thomas Lüthi, improving to 17th in the standings with 23 points.
In 2007, with the team renamed Emmi – Caffè Latte and switching to Aprilia, Cortese broke into the top ten regularly and finished the championship fourteenth. For 2008 he rode works Aprilia RSA 125 machinery and, despite a difficult opening half of the season, finished consistently in the top eight from mid-season onward. He was the only 125cc rider to complete every race that year, scoring points in all but one.
At Ajo Motorsport in 2009 on factory Derbi RSA 125 bikes — the same squad that had guided Mike Di Meglio to the 2008 title — Cortese reached the podium three times and finished sixth in the championship. In 2010 he was teammates with Marc Márquez and Adrián Martín at Ajo, taking two podiums at the Sachsenring and Indianapolis and achieving pole position in Italy to finish seventh overall.
Moving to Dirk Heidolf's Racing Team Germany for 2011, Cortese took his maiden Grand Prix victory — his 109th career start — in the Czech Republic, having battled Johann Zarco in the closing laps. He ended the season fourth in the standings.
The 2012 season, contested under the new Moto3 regulations, brought Cortese his first world championship. He claimed the Moto3 title with Intact GP, having contested every Grand Prix from his 2005 debut through to the 2016 French Grand Prix — a remarkable streak of consistency broken only by a knee injury that forced him out of that race.
Cortese stepped up to Moto2 in 2013 with Dynavolt Intact GP. His first podium in the intermediate class came at Brno in 2014. He remained with the team through the end of the 2017 season without securing a full campaign for 2018.
After failing to find a Moto2 seat for 2018, Cortese signed with Kallio Racing to compete in the Supersport World Championship on a Yamaha YZF-R6. In his debut season in the category, he won two races — at Aragon and Donington Park — and secured eight podium finishes in total, claiming the 2018 Supersport World Championship title. The achievement was all the more notable for coming in his first year in the series.
Cortese moved to the Superbike World Championship in 2019 with the GRT Yamaha World SBK team, finishing twelfth in the standings. For 2020 he transferred to Team Pedercini, switching from Yamaha to Kawasaki. At the Portimão round he suffered a heavy crash in the first race, sustaining multiple injuries that required eight months of recovery. He did not race again in that season.
Nearly three years after the Portimão crash, Cortese announced his retirement from motorcycle racing via his Instagram account. He credited local emergency medical teams with preventing paraplegia but stated that his body had never fully recovered. He subsequently began working for one of his former sponsors, Gutmann Gruppe, and lives in Berkheim.