Azur started his career as a sportswriter for Hadashot.
Azur is the owner of advertising rights for Israel Plus. With Pini Zahavi, he also runs Charlton, a company which owns broadcasting rights for football matches in Israel.
Through Mirkaei Tikshoret, he runs radio stations in Israel and The Jerusalem Post. Canadian investor Leonard Asper sued for a controlling stake in the Post in 2005 after failed negotiations with Azur. The lawsuit was dismissed in 2006. In 2012, he launched Sof Hashavua, a newspaper only published on Fridays. In 2014, he acquired Maariv, another Israeli newspaper.