I agree with Harp, I think your turkey is sick. Maybe the move stressed him out... maybe he came into contact with something that made him unwell?? I would possibly move him back right away and try to figure out what could be changed with his new coop.
Also, are you giving him HIGH-Protein feed, such as game bird starter? 24-26% is VERY important for their health. Feed hard boiled egg yolks for extra protein if you're in a pinch. If he hasn't had enough protein, any stresses could upset his system very easily. He should also continue having the heat lamp, since he's 3 weeks old, that is pretty young. Turkey poults are nowhere near as hardy as chicks until their 8 -9 weeks old.
I don't use medicated feed or antibiotics at my place, but others might recommend something like that too.
Keep us posted.