I assume that the “European migration theory” you mention here is the same as the Solutrean hypothesis. If so, this theory does not completely challenge the land bridge hypothesis. It does not challenge the idea that people came to America across the land bridge. However, it does argue that some of the earliest Americans could have come from the European people known as the Solutreans.
According to the land bridge theory, all of the natives of the Americas came to the land mass across the land bridge where the Bering Strait now exists. Ice ages locked water up in ice caps, making sea levels drop. This created a land bridge where there is now relatively shallow water in the Bering Strait.
The Solutrean hypothesis acknowledges that some people came to the Americas in this way. However, it also argues that some people came to the Americas from Europe. It argues that these people came across the North Atlantic at a time when pack ice reached from Europe to the Americas. It holds that these people had technology and skills like those of modern Inuits. These skills and technology allowed them to paddle along the pack ice, eventually reaching the Americas.
This theory challenges the land bridge theory because it says that not all of the natives of the Americas came across the land bridge from Asia.