The shocking footage shows the van driving at least twice around a roundabout leaving skid marks on the road as it screeched around.
Yodel said the driver had been identified as a supplier and “immediately removed from our business”.
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A spokesman said: “This behaviour is completely unacceptable and we are passing his details to the police.
“Yodel takes the safety of its workforce and the general public extremely seriously. All of our drivers are provided with training and strict codes of conduct which they are expected to adhere to at all times.”