It takes certain developments in shipbuilding to cross oceans reliably, particularly the need to build ships that can take storms and high waves well. Crossing the North Atlantic without a keeled ship is a good way to guarantee shipwreck.

There's no habitation of intermediate island hopping points that one might expect from transoceanic-capable cultures before European exploration: islands like the Falklands, Bermuda, Azores, Iceland. It's plausible that isolated, shipwrecked people drifted to land via oceanic currents, but that's not the kind of contact posited.

Viking precolumbian contact is confirmed. Polynesian contact is suspected, but the only evidence for this is the sweet potato, and there is a maddening lack of cultural, linguistic, or archaeological evidence to back this up. Pretty much every other hypothesis tends to be rooted in individual misunderstanding of evidence, very loose artistic correspondence ("this image from <insert culture here> totally looks like <insert other culture here>. CONTACT CONFIRMED!"), nationalistic priority claims (particularly of the discovery of Grand Banks, and Chinese claims to have gotten to the Americas before Columbus), or outright fraud.