One consideration is the Barring straight is very narrow compared to other ocean passages, so people may have regularly crossed each side without thinking 'new land'!
My whole point is that you don't know who "their" is. Please stop and think about who you mean when you use the word "their". Do you mean 1 person, or 2 people, or a 100, or a million? And the map you are linking to did not exist 20,000 years ago!
Their being the people who crossed from one side to another.
1 or 100,000 the point is people who made the crossing did not get a popup saying "achievement unlocked: new continent!" And then run around telling everyone about it.
Yes, European mapmakers did not known about such crossings, but being a very remote area in 1400's they probably never asked. And from an oral history perspective it would not be anything unusual because a 50 mile ocean crossing is just a regular thing to sea going populations.
Also, DNA evidence is going to be sketchy because it's really hard to say if someone migrated 400 years ago vs 600 years ago.
PS: I am not saying this 100% happened, but it seems very likely.
From their perspective the map looks like this: https://en.m.wikipedia.org/wiki/Circumpolar_peoples
From their perspective North America is simply yet another settled area.