After graduating high school in 1955, Norton initially went to Oakland City College staying for just one year. He left in 1956 for San Jose State College, where he was coached by Lloyd (Bud) Winter. He first achieved national fame by equalling the world record of 9.3 for 100 y as a college junior, in San Jose on 12 April 1958. The next year, 1959, was an outstanding one. He won three gold medals at the 1959 Pan American Games and he tied Leamon King's record at the 100 m at 10.1 s in San Jose on 18 April. His achievements in 1959 were recognised by being voted Track and Field News's United States Men's Athlete of the Year - the inaugural award of this honour.
In 1960, Norton carried on his impressive form of the previous year by tying four world records: he equalled the 220 y record of 20.6 s in Berkeley on 19 March; equalled again the 100 y record of 9.3 s in San Jose on 2 April; equalled the 200 m record of 20.6 s in Philadelphia on 30 April; and equalled the newly set record for the 200 m of 20.5 s in Stanford on 2 July. He qualified for the 100 and 200 at the 1960 Olympics by coming first in both events at the United States Olympic Trials, equalling the world record in the process in the 200 m. However, his form at the Olympics itself deserted him, most probably because of nerves, and he finished a disappointing last in both the 100 and 200 m finals. Norton's failure on Thursday 1st September in the 100 m was one of such disasters that befell American favorites that day and the day was to become known as 'Black Thursday'. He tried to make amends for his failures in the individual events in the 4x100 m relay. The team of Frank Budd, Norton, Stone Johnson and Dave Sime finished first but were disqualified because Norton had started his run too early and his exchange from Budd happened outside the exchange zone. Norton came to the attention of the world's press at the Olympics for more than his athletic tribulations when he was seen courting his fellow American sprint star and public favourite Wilma Rudolph.
Early in 1960, Norton flunked out of college and the social work degree he was working on. Norton continued to train at the San Clara Valley Youth Village. Norton retired from athletics that season and was drafted as an American football player at the end of that year.
In 1966, Norton sought the nomination to sit as a Republican for the California Assembly 17th District (Oakland and Berkeley). At the time he was working as a partner in a service agency for attorneys.