American English would only say "I have the sheet" not "the bed sheet". If it needed to be clearer the language would be "I have the sheet for the bed under my chair." Bed sheet would be understood but viewed awkward speach.
I disagree. I would definitely say "the bed sheet" before I would say "the sheet for the bed," if I needed to be clear I wasn't talking about a sheet of paper, for example. The second formulation is the awkward one.