Conway was coached mainly by Carlo Fassi and also spent a summer in the United States training under Robin Cousins. She would win six British national titles, claiming her first at the age of 14. She made her major championship debut at the 1986 European Championships, finishing 11th. She went on to achieve top-ten placings three times at the World Championships and four times at European Championships. Her most successful year was 1991, when she finished 4th at the European Championships and 7th at the World Championships. She also competed twice at the Winter Olympics, finishing 12th in 1988 and 18th in 1992. She was the first British woman to land a triple flip successfully in competition, at the 1991 European Championships.