The public understanding of climate change plays a critical role in translating climate science into climate action. In the public discourse, climate impacts are often discussed in the context of extreme weather events. Here, we analyse 65 million Twitter posts and 240 thousand news media articles related to 18 major hurricanes from 2010 to 2022 to clarify how hurricanes impact the public discussion around climate change. First, we analyse news content and show that climate change is the most prominent non-hurricane specific topic discussed by the news media in relation to hurricanes. Second, we perform a comparative analysis between reliable and questionable news media outlets, finding that the language around climate change varies between news media providers. Finally, using geolocated data, we show that accounts in regions affected by hurricanes discuss climate change at a significantly higher rate than accounts in unaffected areas, with references to climate change increasing by, on average, 80% after impact, and up to 200% for the largest hurricanes. Our findings demonstrate how hurricanes have a key impact on the public awareness of climate change.