Puffins and razorbills rest on land in Britain but winter in the mid-Atlantic. Some birds can ride out a storm pretty well using the wind. However, they cannot stay in the fixed vicinity of a rig at such times, so you wouldn't see them during storms. In stormy weather, flocks of gulls would sit on our wall and roof and scavenge from our dustbins, while nearby rubbish tips also had seagulls by the thousand. These apparently conflicting answers are all partly correct because what happens depends on the species, their location and the weather conditions.
Some birds can sit out storms on the water surface.
The auks, which include the puffins and guillemots, leave land at the end of the breeding season and are adapted to living at sea. During their moult, puffins temporarily lose the ability to fly and have little choice but to stay on the water. Fortunately, because they are diving birds, temporary submergence in rough weather is not going to trouble them. Other diving birds, including ocean-going penguin species in the southern hemisphere, can also sit out storms. Larger species may take to the air.