How do small birds survive the winter?
There are a variety of physiological, anatomical and behavioral adaptations that keep birds warm in winter.

For starters, birds are endothermic ("warm-blooded"), similar to mammals. You suggested that the coats humans wear are superior to feathers as insulation. Yet some of the finest coats and sleeping bags were long filled with feathers (e.g. goose down). Animals that live in cold regions tend to have smaller extremities. For example, you probably won't find many long-legged flamingos or ibises in the Arctic during the winter (though food supply is another reason they couldn't survive there).

As someone else suggested, water temperature is a relative constant, which means water (or better yet, snow) can actually insulate animals from colder temperatures.

Countercurrent heat exchange is a strategy that helps endothermic animals conserve heat. The page I linked to is part of a book titled Animal Adaptations to Cold, which includes a chapter titled "Avian [Bird] Adjustments to Cold."