Just to clarify
Erasing the history on your PC will make no difference, if they were checking they would look at your server/ firewall.
As someone else said they would normally only trace something that had been flagged up by the firewall because it is a banned site or because that site is seeing unusual or high usage (as a whole or by an individual - they can see this if they want to).
Really popular sites like Yootube and eBay might get flagged up because they could indicate non work related usage.
BBC is unlikely to be flagged becuse it is seen as news/educational and I would be surprised if PC got flagged except possibly as a forum.
I have never worked doing this in an IT department but I worked with lots of IT departments that did it in my previous BT employment.