Fushida's driving career began in 1965 as a factory driver for Honda. In 1966 he joined Toyota and was among five drivers who participated in the Toyota 2000GT Speed Trial at the Yatabe High Speed Test Track. Racing in Japan through the late 1960s and into the 1970s, he won the Fuji 1000km endurance race four times — in 1968, 1969, 1971, and 1979 — and the Suzuka 1000km twice, in 1968 and 1971. He also won the 1972 Fuji Grand Championship Race title and competed in the 1968 and 1969 Japan Grand Prix sports car races.
After leaving Toyota following the 1969 season, Fushida moved to the United States to compete in the SCCA Continental Championship, the Canadian-American Challenge Cup, and the Trans-Am Series. His American campaign ended when he sustained injuries in a Trans-Am accident at Road America in 1971, after which he returned to Japan.
In 1973, Fushida and fellow Japanese driver Tetsu Ikuzawa became the first Japanese competitors to race at Le Mans, sharing the Mazda rotary-powered Sigma MC73. Fushida would make two further Le Mans entries; his final appearance came in 1981 for Mazdaspeed. All three of his Le Mans starts as a driver ended in retirement.
His Formula One career was brief and did not reach the starting grid. Entered by the Japanese Maki team, Fushida attempted two Grands Prix in 1975: at Zandvoort in the Netherlands a blown engine prevented him from starting, and at the British Grand Prix at Silverstone he failed to qualify. He was subsequently replaced at Maki by Tony Trimmer and never again entered a Formula One event. Despite failing to start either race, his entry at Zandvoort secured his place in history as the first Japanese driver to enter a World Championship Formula One Grand Prix.
In a separate result that year, Fushida and co-driver Don Holland finished first in Class C and fifth overall at the 1975 Hardie Ferodo 1000 at Mount Panorama Circuit.
After retiring from racing in 1981, Fushida transitioned into motorsport management. He worked for Dome Racing from 1986, then joined the TOM'S tuning company in Japan in 1989. In 1992 he moved to the United Kingdom to oversee TOM'S entry into British Formula Three as TOM's GB. In 1998 TOM's GB was acquired by Audi and became Racing Technology Norfolk (RTN). As operations director of RTN, Fushida oversaw the Bentley Speed 8's winning campaign at the 2003 24 Hours of Le Mans — returning to the venue where he had made history as a driver thirty years earlier, this time as part of the winning effort.
Following the Le Mans triumph he returned to Japan, where he joined Dome and eventually succeeded founder Minoru Hayashi as company president in September 2012, serving until his retirement in July 2015. He subsequently worked as a consultant for Dome.