Why do shallow straits such as the Bering Strait now separate lands that were joined during the last ice age?

Answer: The sea rose and flooded the low land between them

The sea rose and flooded the low land between them ✓ — At the peak of the last ice age so much water was locked up in ice sheets that the sea stood about 130 meters lower than today. The Bering Strait is only about 50 meters deep, so its floor was a broad, dry plain linking Siberia and Alaska; as the ice melted, the rising sea covered it between roughly 13,000 and 11,000 years ago. Many of the world's shallow straits were flooded in the same way.

Rivers cut channels through the land that joined them — Rivers did not cut these straits. Rivers carve valleys that run downhill to the coast, and the Bering land bridge disappeared because the sea rose over it, not because anything sliced through it. When the sea drowns a river valley, the result is a long inlet such as Chesapeake Bay, not a passage between two seas.

Moving tectonic plates pulled the joined lands apart — Plates move far too slowly for this: at a few centimeters a year, about the speed fingernails grow, they have shifted a kilometer or two at most since the ice age, while the Bering Strait is about 80 kilometers wide. Alaska and Siberia also still sit on one shallow shelf of continental crust, which is simply under water now. Plate motion does open seaways, but it takes millions of years.