Turner was Founding Director of London Design Biennale in 2016, and the Artistic Director in 2018. He was the former Editor of Icon Magazine and Modern Painters. 

Turner studied anthropology, archaeology art history at the University of Cambridge and has a PhD from the University of London. He is an editor of Cabinet magazine and has curated exhibitions at the Venice Architecture Biennale, Manifesta 7: the European Biennial of Contemporary Art and the Arnolfini Gallery, and writes for the London Review of Books, The Guardian, Apollo and Sunday Telegraph.