Depending on the situation, anything from a scrap to a sheet could be a “bit”.
Northern Californian.
Another Brit - if I asked someone with a pile of A4 sheets in front of them for a ‘bit’ of paper I’d expect a whole sheet; if desperate enough to ask someone with only one sheet in front of them I’d expect them to fold and rip me off a portion; in a pub asking the bartender for a ‘bit’ of paper I’d expect anything to write on - a bit of till roll, a beer mat, a page from a notebook; asking beloved husband at home I’d expcet him to pass the nearest paper like product to hand regardless of size, shape or quality (old envelope, post-it note, old photocopy from my job etc.). As with most questions of language usage the context is all.