CAIRO — Tensions between Iran and Saudi Arabia deepened on Thursday as Iranian leaders lashed out with rare vehemence against the continuing Saudi air campaign in Yemen, even hurling personal insults at the young Saudi prince who is leading the fight.

Ayatollah Ali Khamenei, the supreme leader of Iran, on Thursday denounced the Saudi airstrikes in Yemen as “a crime” and “a genocide,” while all but taunting Saudi Arabia that its war in Yemen was doomed to fail.

A regional coalition led by Saudi Arabia extended its bombing campaign for a 16th night in its effort to stop the Houthi movement and its allies from dominating Yemen. The Houthis nonetheless continued their advance, and aid groups warned of a compounding humanitarian catastrophe, particularly in the port city of Aden.

Secretary of State John Kerry sharply warned Iran over its backing for the other side of the conflict in Yemen, in the first explicit American accusation that Tehran has been providing military aid to the Houthis.