it would not be a land bridge if made of ice. The Bering Sea is relatively shallow (Alaska and eastern Siberia are part of the same geological continental land mass or plate, much like England is part of Europe), and like the North Sea and English Channel, the Bering sea was above sea level during peak glaciation when ice accumulation on land (on the continents) resulted in sea level being several hundred feet lower than today.
So, just as animals and humans could walk from mainland Europe to The British Isles, animals and humans could walk from Asia to North America. The open land region was somewhat restricted in the Alaska/Pacific Coast region, though, primarily because the mountains in western North America are almost right up against the Pacific Ocean, and mountains were covered by glaciers too. There was, apparently and intermittently (not permanent), an ice-free corridor on the interior, between mountain glaciers and the main North American ice mass, which was one of the two routes that animal life could use to migrate. Archeological investigations suggest that the coastal route was also used by humans. Not all migration was along the inland corridor as some have thought for a long time.