Garozzo won the 2008 Cadet World Championships at home in Acireale, then took a silver medal in the 2011 and 2012 Junior World Championships. In the 2013 Summer Universiade he was defeated in the semifinals by Russia's Aleksey Cheremisinov and came away with a bronze medal. In the 2014–15 season he climbed his first World Cup podium with a silver medal in the Challenge International de Paris. He earned a silver medal at the 2015 European Championships after being defeated in the final by teammate Andrea Cassarà.