The world-bearing turtle is real, widespread and frequently misattributed, and the specific traditions are worth separating.
Haudenosaunee and Anishinaabe. In Haudenosaunee accounts, Sky Woman falls from the world above and is caught by water birds and set on the back of a great turtle; animals dive for mud, and the muskrat, the smallest and least likely, succeeds at the cost of its life. The earth grows on the turtle's back. Anishinaabe earth-diver accounts are closely related. Turtle Island as a name for North America comes from these traditions, is used by many Indigenous nations, and has been adopted more widely, including in activist and environmental contexts. It is not a generic Native American term but a specific one with a specific source.
Hinduism. Kurma is the tortoise avatar of Vishnu. During the churning of the ocean of milk, the gods and asuras use Mount Mandara as a churning stick, and when it begins to sink Vishnu takes tortoise form and supports it on his back. The episode appears in the Mahabharata and the Puranas.
China. The black tortoise Xuanwu is the guardian of the north and one of the four symbols of the sky, typically depicted entwined with a snake. The tortoise also carries stelae in Chinese monumental sculpture, the bixi, a stone tortoise base for inscribed tablets, is one of the most common structures in Chinese temple and tomb complexes.
Divination. Turtle plastrons, alongside ox scapulae, were the medium of Shang dynasty pyromancy in the second millennium BCE. Heat was applied, the cracks were read, and the question and answer were inscribed on the shell. Those inscriptions are the earliest substantial body of Chinese writing that survives, so the turtle is quite literally the surface on which the written language first appears to us.