Do domestic turkeys drown in a rainstorm?

Ordinary heavy rain won't actually drown a turkey, but they will get wet and miserable if they don't have enough dry shelter. If the rainstorm actually causes flooding then yes, turkeys (and chickens too, for that matter) can drown, because they can't fly and their feathers get waterlogged, making movement difficult for them. They should always be protected from flooding if they're kept in areas where this is a risk, with their runs and sheds built on high ground with good water run-off, and rescue of them made a priority in the event of flood.