Avcı was born in Larnaca in 1959. After graduating from Famagusta Namık Kemal High School he studied Electrical and Electronics Engineering at American University of Beirut, graduating in 1985. He received his master's degree in Business Administration at the Eastern Mediterranean University (EMU) in 1992 and has a doctorate from Çukurova University on Administration and Organization.
He attended trainings in USA and Germany in the fields of tourism, administration and quality.
He was a Member of North Cyprus Atatürk Research and Implementation Centre and Member of the State Television (BRT) Board. In the public sector, he served as a Member of the Advisory Council, Specialization Commission, Tourism Master Plan Working Committee, Tourism Advisory Committee, and Human Resource and Education Committee. In the 14 December 2003 General Elections and later in the 20 February 2005 early elections he was elected to the Parliament as National Unity Party's Famagusta MP. After serving as the Secretary General of the National Unity Party, he resigned from his party and formed the Freedom and Reform Party. He took part in the Republican Turkish Party-United Forces and Freedom and Reform Party coalition government, which was formed on 25 September 2006, as the Deputy Prime Minister and Minister of Foreign Affairs.
As Minister of Foreign Affairs, Turgay Avci has begun some very good work with the embargoes. He has begun to directly ship tours and to normalize relations with Syria.