Would those crossing the Bering Strait already have had these weapons? And wouldn't they have changed in all that time, on a separate trajectory (pun intended!) from the European weaponry?
The bow and arrow was independently invented in the New World (about 10kya last time I checked; see Blitz, John H. "Adoption of the bow in prehistoric North America." North American Archaeologist 9.2 (1988): 123-145.)
Both groups (Native North Americans and Europeans) were following somewhat similar trajectories in overall archery (which is quite a bit older in the Old World). No recursive bows or crossbows in the New World (and I don't think there were any true longbows either, but I could be wrong, my citation on that is from 1992...Hardy, Robert. Longbow: a social and military history. Bois d'Arc Press, 1992.)
Should be pointed out that there was no one single pre-Columbian Native bow. There were wide ranging designs; From Really big bows in the Amazon that shoot giant 5 ft arrows, to little short flat-bows along the Pacific coast, to composite hide bows out on the planes, to cable backed construction up in the arctic. The different tribes in the different areas all had their own take on how to build a bow.