Hi there, while there's always more to be said on this topic, perhaps this post from our FAQs can be helpful to you.

I imagine /r/AskAnthropology would be a better place.

Well, the land bridge clearly existed and people clearly passed from Siberia to North America. But the details are in debate. Previously, the main theory was "Clovis First" which is your classic land bridge theory. Big game hunters walk across and ice-free corridor from Asia into America, and explode across the landscape. They use a particular kind of stone spearpoint, and are known as the Clovis culture.

...Only more recently, evidence for humans in the Americas before Clovis, including way down at the tip of South America, has been found. But it's thought there wasn't an ice free corridor across Canada at this point.

So a common theory is that these people followed the coast from Beringia (the land bridge was still there, after all) all the way down to Chile, and dispersed across the Americas before developing into Clovis.

There's also the Solutrean hypothesis, which claims America was populated by Europe, but there's no real support for this one.

Relevant link

http://www.nature.com/news/ancient-migration-coming-to-america-1.10562