That is actually under debate.
Both the time and how they went there.
As that land bridge: actually did not exist at that time.
It was a gigantic pillar of ice.
Does not mean it was impossible to pass ... but to get there: you already have to pass thousands of miles of icy land, where nothing lived at all.
So: it might have sounded plausible when the books saying this were written, it is absolutely not plausible from a practical point of view.
Knowing that the oldest artefacts from "early settlers" are over 40k years old, we have proof the time frame is wrong.
How did the people got there? Most likely by boat. How else? While the idea that some people walked from Africa to China, to Siberia, crossed a land bridge to Alaska, walked down north America, to finally reach South America is not absurdly wrong, but it is far less plausible than boats from China to West Coast America.
Why do I write this "nonsense"? Well a few month ago I read an article who they think Japan was settle, roughly 20k years ago. They tried to row there from Taiwan, I believe. In sea worthy rowing boats. They managed the trip in 30 days, an concluded: it was possible for early settlers to travel by boat to Japan, roughly 20k or 30k years ago.
That is the biggest nonsense ever: because of the ice mountains over the land bridge connecting north America with Siberia: Japan was connected by land to China. You just walk there. Or tame some Mammoths and ride on them ... there was perhaps a large lake between China and Japan, but the land masses were connects.
Except for a small gab, which you probably could look over to the other side: everything from China down to Australia was connected, because the sea level was close to 200m lower than right now (yes, it is not 200 ... but close enough for a rough number)
Same stupidity about "how was Australia settled". What we call Aboriginals settled there roughly 40k years ago. And the mystery always was sold as: how did they get there? No one knows!! Yeah, dear IQ monsters: it was in the middle of an "ice age". Actually at the peak of the last ice age. Except for the last 30km ... they walked there. So much to "land bridge".