[1] He began his football career as an apprentice with Brighton & Hove Albion in 1972, and made his senior debut in the last match of the 1972–73 Second Division season, at home to Nottingham Forest on 28 April 1973.
[2] At the age of 17 years, 48 days, he became the club's youngest league debutant – a record he held until 1987 when the 16-year-old Ian Chapman first appeared[2][3] – and scored the equalising goal, albeit via a deflection.
[1][2] Conway turned professional with Brighton in 1974, but played only once more for the team before joining Fourth Division club Swansea City in December 1975 for a £3,000 fee.
[2] Conway soon established himself in the first team and, according to a profile on Swansea City's website, "became a favourite with his pace and sparkling control on the left wing.
He suffered serious injuries in a car crash,[6] attempted a comeback against Barnsley in April 1978, and during the match was rushed to hospital with liver damage, believed to have been a consequence of the accident.