WASHINGTON—President Donald Trump ousted chief strategist Steve Bannon on Friday, as newly minted Chief of Staff John Kelly sought to bring order to an administration riven by infighting and power struggles, and increasingly at odds with congressional leaders.

Mr. Bannon’s departure marked the fourth senior official in five weeks—and sixth in seven months—to leave the Trump administration, which has been unable to take advantage of a Republican-controlled Congress to push through its legislative agenda.

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