As Republicans tanked their own bid to impeach Alejandro N. Mayorkas, the homeland security secretary, Mr. Johnson, who minutes before had been buttonholing holdouts on the House floor, was the face of the failure, a slightly panicked look on his face and his cheeks flushed as he announced the loss.

Then the House moved on to a second vote Mr. Johnson had orchestrated, on a $17.6 billion aid package for Israel that he knew would not muster the votes it needed to pass.

It also failed.

The back-to-back defeats highlighted the litany of problems Mr. Johnson inherited the day he was elected speaker and his inexperience in the position, roughly 100 days after being catapulted from the rank and file to the top job in the House. Saddled with a razor-thin margin of control, and a deeply divided conference that has proved repeatedly to be a majority in name only, he has struggled to corral his unruly colleagues and made a series of decisions that only added to his own challenges.

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