The Bering Land Bridge comes up in archeological discussions of how the Indians came to America. Geological changes are thought to have raised the land of the Bering Straits above sea level allowing the ancestors of the Indians to walk across from Asia. The main geology evidence in favor is that the modern Bering Straits are fairly shallow. If there is any other evidence in favor of an ancient land bridge, I never heard it.
Actually, a land bridge is not needed for the Indians to cross the Bering Straits. Modern Alaska Eskimos cross the straits, in skin boats. Or at least they did up until the late 1940's when the Soviets made life difficult for any American Eskimos found on their side of the straits. Which generated a strong anti-communist spirit among the Eskimos that persists to this day.
The Eskimo's used skin boats, umiaks. While this sounds primitive, the skin was walrus hide, quarter of an inch thick and tough as fiberglass. They were built up to 30 feet long, and were strong enough to withstand the thrust of a 40 horsepower outboard motor. Evinrude was popular among the Eskimo, more so than Mercury or Johnson.
And, if you have ever traveled in the back country, you know that canoe is the way to go. Given a water way, two men and a canoe can carry a thousand pounds of cargo, and do it faster than walking pace. Whereas afoot the same two men can only backpack a hundred pounds of food and gear. The Indian canoe is so good a water craft that it is still in production today. Granted modern materials give a vaster stronger vessel, but birchbark is strong enough to make a useful canoe. The crew must take greater care in the rapids, touch a rock and you have a big leak, but it is perfectly do able.
So I have no trouble believing the Indians followed the coast line, and paddled across the open strait in good weather. The only timing issue I see is the state of the Ice Age. At the height of glaciation,10,000 years ago, North America must have been as bleak as the North Pole. But when the glaciers melt back, the grass comes up and game animals appear to eat the grass then enterprising bands of Indians or proto-Indians could have emigrated to the New World. Archeological finds in North America are mostly dated after the last Ice Age. There are a few sites claimed to be earlier, but not many, and the pre glacial sites are "controversial", i.e. the archeologists still argue about the dating.