In 1974 in space of four days he broke Mark Spitz's 200 m freestyle world record, Rick Demont's 400 m freestyle world record and Steve Holland's 1500 m freestyle world record.
He appeared on the cover of Sports Illustrated magazine on August 4, 1975, after receiving the FINA Prize Eminence Award in 1974, representative of the greatest contribution to world aquatics; being named World Swimmer of the Year for two years running, 1974 and 1975; and winning the Sullivan Award in 1975 as the most outstanding amateur athlete in the United States after having won three gold medals at the World Aquatics Championships.
In the 1976 Olympics the following summer, he could manage only a silver medal in the 400 m freestyle; he was completely shut out of his signature event, the 1500 m freestyle. He could not compete four years later when the US led the 1980 Summer Olympics boycott in Moscow, Russia. Shaw further was a member of the 1984 US water polo team that was undefeated in Olympic competition but was awarded the silver medal because the Yugoslav team, with an identical record, scored four more goals overall.