LONDON — A recent semi-scandal over seats on a train illuminates the deep problems of Britain’s Labour Party. Its left-wing leader, Jeremy Corbyn, made a video sitting on the floor of a train, calling for the renationalization of the railways while claiming that there were no empty seats and that travelers were “ram-packed.”

Virgin Trains, which operates the service, then released closed-circuit television images showing numerous vacant seats bypassed by Mr. Corbyn, who was later filmed in a seat for most of the journey.

That many of Britain’s privatized railways are badly overcrowded, especially in southern England, is indisputable. But the episode was another indication that the Labour Party is in shambles: Its leader and its members of Parliament are in a virtual civil war, and it is deeply unpopular with the broader electorate.

Yet the party’s membership is soaring. Under Mr. Corbyn, some 300,000 people have joined Labour in the last year. Labour has more members, about 500,000, than all the other British political parties put together.