Can and his twin brother Deniz Öncü both grew up competing in junior motorcycle racing, often running side by side in the same championships. The brothers competed in the Asia Talent Cup in 2016 and 2017, and both entered the Red Bull MotoGP Rookies Cup in 2017. Can finished third that year and went on to win the Red Bull MotoGP Rookies Cup in 2018 while Deniz finished second. Both brothers are mentored by Kenan Sofuoğlu, the Turkish multi-world champion motorcycle racer.
In 2018, alongside his Rookies Cup duties, Can joined Ajo Motorsport for the FIM CEV Moto3 Junior World Championship. Later that season, he was entered as a wildcard in the Moto3 World Championship at Valencia. He capitalised on the opportunity to become the first rider to win on their Grand Prix debut since Noboru Ueda in 1991, and broke the record for youngest GP race winner previously held for ten years by Scott Redding. His entry was made possible by a 2018 FIM regulation change that allowed the reigning Red Bull Rookies Cup champion to compete in Moto3 at a minimum age of 15, rather than the standard requirement of 16.
For 2019, Öncü made his first full Moto3 season with Red Bull KTM Ajo. Despite the promise shown through his wildcard victory and Rookies Cup title, he struggled to replicate that form across a full campaign. He scored only three points finishes from sixteen contested races and ended the season with eight points and thirty-first position in the championship. He was dropped from the Ajo team at the end of the year.
From 2020, Öncü moved to the Supersport World Championship with the Turkish Racing Team — a squad headed by his mentor Kenan Sofuoğlu and operating in partnership with Puccetti Kawasaki and Orelac Racing. He collected regular points finishes in his first full WorldSSP season, ending the year twelfth overall. In 2021 he improved to regular top-ten form and took his first WorldSSP podium at the French round. He remained with the Turkish Racing Team through subsequent seasons, making 2024 his fifth consecutive year with the outfit.
Öncü also made wildcard entries in the CIV Supersport 600 Championship in 2021 at Mugello, using the Italian domestic rounds to accumulate additional seat time.
The 2018 Valencian GP victory stands as the defining moment of Öncü's career to date. His debut win placed him in the record books alongside the great GP debut winners of history, and it sparked significant expectations for a rider who was still months from his sixteenth birthday. That he achieved the feat on machinery prepared by Ajo Motorsport — one of the most respected junior GP teams — lent additional credibility to the performance.
Can Öncü's historic 2018 victory made him a landmark figure in Turkish motorsport alongside his brother and mentor. While his full-time Moto3 campaign did not build on that debut result, his transition to WorldSSP established him as a durable international competitor. He remains, along with Deniz, one of the most visible representatives of an emerging Turkish presence in the world motorcycle racing paddock.