To be clear, there's evidence of Euros, but it's mixed and rare. If they did come, most died out or left. As the link below notes, the earliest sign of
solid evidence is Norse from a 1000 years ago:
http://www.let.rug.nl/usa/outlines/history-1994/early-america/the-first-europeans.php
The first Europeans to arrive in North America -- at least the first for whom there is solid evidence -- were Norse, traveling west from Greenland, where Erik the Red had founded a settlement around the year 985. In 1001 his son Leif is thought to have explored the northeast coast of what is now Canada and spent at least one winter there.
There's also indications of Polynesian immigration in South America.
https://www.nature.com/articles/d41586-020-01983-5
DNA analysis of Polynesians and Native South Americans has revealed an ancient genetic signature that resolves a long-running debate over Polynesian origins and early contacts between the two populations.
Most evidence indicates American migrations were over the Bering land bridge during Ice Ages up to 20,000 years ago as you stated.
I don't see what the problem is here. All human beings originated in Africa and spread from there in waves. It's of academic interest to see how humanity spread across the planet, not a sports competition much less a racist competition.