Last winter after a heavy snow I had a couple of problems with ice
dams. One above the exhaust hood over the stove, another above the
bathroom. Both appear to be caused by improper installation of the
fans, they both vent into the attic space. I'm going to correct this
by having them vent through the roof.

My question is this, wont the warm air coming out of those vents melt
the snow on the roof, and wont the resulting water end up freezing
into an ice dam as it tries to run down the cold roof where it runs
into more snow?