24 people with minor injuries have been taken to St Thomas' hospital, a London Ambulance Service spokesperson has said.
Several others in a more serious condition are being treated at the scene of the crash.
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Dozens of women gathered on Westminster Bridge to form a human chain in a show of solidarity with the victims of Wednesday's terror attack.
In a wide-ranging Peston on Sunday interview, he said Parliament security needs reviewing and the PM's counter-terror strategy is flawed.
The terror attack carried out by Khalid Masood at Westminster that left four people dead lasted less than two minutes say police.