Trump says Iran war moving 'very rapidly'

US President Donald Trump said Thursday the war against Iran was moving "very rapidly," even as Tehran's new leader vowed defiance in his first public message.

"The situation with Iran is moving along very rapidly. It's doing very well, our military is unsurpassed," Trump said at the White House.

"They really are a nation of terror and hate, and they're paying a big price right now," added the US leader, who was attending a Women's History Month event with First Lady Melania Trump.

Trump has given mixed signals in recent days about the progress of the war, saying in recent days that "we won" against Iran and that it could end "very soon", but also insisting on the need to fight on.

He did not directly respond to comments by Iran's new supreme leader Mojtaba Khamenei, in his first message since his elevation to the post, vowing revenge and saying that Iran must keep a chokehold on the Strait of Hormuz.

Global oil prices have soared with the virtual halt of tanker traffic through the strategic waterway.

But Trump said on social media earlier Thursday that stopping Iran's "evil empire" from getting nuclear weapons was more important.