There is some evidence for a land bridge having existed, but the question itself is considered "white mans foolishness" by natives in the area.

Their answer would be "who cares". The sea freezes over every winter anyway, and people routinely walk from Nome to Siberia every year. Russian authorities have turned back many such folks in recent years.

The concept of a land bridge was invented out of necessity to explain the apparent migration of people and animals from Asia to Alaska. No such bridge is necessary to explain it. Even today, they just walk across in the winter. The "scientists" apparently all live in California, where no such thing would ever be thought of.

The above are just plain wrong. The Bering Strait does not freeze over, and it is not possible to easily walk from Alaska to Siberia. At best, even before the current warming phase has broken up Arctic icepacks, the only way to walk between the two would be across the Arctic icecap. While one would not have to go all the way to the North Pole, crossing would entail a very dangerous journey of several hundred miles, at the minimum.

Crossing from Asia to the Americas over the Arctic ice is so long and dangerous, and has been for tens of thousands of years, that it is practically impossible. Certainly, it cannot account for the movement of animals and men from Asia to the Americas.

Back to the question: the evidence for a land bridge is twofold: (1) the topology and depth of the ocean floor in the Bering strait, and (2) geological evidence of significantly lower sea levels at the time in question.

For (1), various sea-floor sampling has shown that significant parts of the ocean floor were actually above water for an extended period of time, around the last Ice Age. The shallow nature of the sea floor also leads to important assumptions about a land bridge.

For (2), we know from the geologic record that sea levels have not remained constant over history, and have fluctuated significantly from current ones. During the Pliestocene age (when the last Ice Ages occurred), the water levels were up to several hundred feet below current. This obviously changes where coastlines are, and exposes significant amounts of "ocean bottom" as actually land.

The question of whether or not there was a Bering Land Bridge is pretty well settled, scientifically. What is still not quite known is the extent that mass migrations made of this land bridge, and the exact time period that it was possible for such migrations to occur.