18th and 19th century Anglo-Americans did come up with some outlandish theories about the origins of Native Americans. Part of this stemmed from the widespread, prejudiced view that non-white people were intellectually and culturally inferior to Europeans and therefore could not have built the great civilizations that European colonizers encountered in the New World.

For example, in an 1885 paper titled "Among the Mound Builders' Remains," Historical and Scientific Society of Manitoba president George Bryce wrote that the Aztecs told the Spanish conquistadors that:

"... before they invaded Mexico from their original home, they were preceded by a civilized race, well acquainted with the arts and science, knowing more art and astronomy in particular than they. They stated that they had exterminated this race."

In North America, ancient Native American structures often took the form of burial mounds. Prior to the 20th century, the majority of scholars (with Thomas Jefferson an interesting exception) claimed that these "Mound Builders" were the descendants of seafaring ancient peoples such as the Vikings, Greeks, Phoenicians, or perhaps one of the "Lost Tribes of Israel." (This last theory was favored by early Mormons.)

Some of the members of Lewis and Clark's expedition (1804-1806) across the Louisiana Purchase territories to the Pacific Ocean encountered Indian tribes whom they believed were descended from Europeans or Israelites.

The idea that certain lighter-skinned or blue-eyed tribes along the Missouri River were descendants of the Welsh were particularly popular during this period. From a letter from Thomas Jefferson to Meriwether Lewis in 1804:

"I inclosed you the map of a Mr. Evans, a Welshman, employed by the Spanish government for that purpose, but whose original object I believe had been to go in search of the Welsh Indians, said to be up the Missouri."

Joseph Whitehouse, one of the expedition members, wrote in his journal about a certain tribe: "we take these savages to be the Welch [Welsh] Indians"

When another expedition member, John Ordway, described Indians with "the Stranges [sic] language of any we have ever yet Seen," he likewise concluded: "we think perhaps that they are the welch Indians."

TL;DR White Americans had a lot of strange theories about Native Americans, based on no evidence but the common idea that non-white people were intellectually inferior to whites.

(Edits: formatting)