A yougov poll suggests otherwise.

"People aren’t stupid, by 3 to 1 more people think the BBC is anti – than pro Brexit, when over a quarter of the audience has come to the conclusion the BBC is biased there is a real problem, because if we don’t want to read the most partisan newspapers we don’t buy them. The BBC’s funding forces us all to pay for it and we expect to be impartial as a result. It is manifestly not perceived as a source of impartial information on this polarising issue."

https://d25d2506sfb94s.cloudfront.net/cumulus_uploads/document/s8s93cmvos/InternalResults_180219_NewspapersBrexit.pdf