What does Hindu scripture say about World Turtle/World Elephant theory?
Here are scriptural sources that describe different kinds of animals supporting the Earth:
World Elephants: In the Bala Kanda of the Ramayana, Indra steals the horse of Ashwamedha Yagna of Rama's ancestor Sagara, and so Sagara tells his sons to dig the Earth to find the horse, as described in this chapter of the Bala Kanda of the Ramayana:
"Let the earth be dug further, may safety be with you, and verily splitting open the surface of the earth you catch hold of the thief of the horse, and on achieving your purpose you may return to me." In this way Sagara ordered his sons. On catching up the words of their father, the great-souled Sagara, his sixty thousand sons rushed towards the surface of the rasaa tala, the netherworld. While digging the earth there they beheld a mountain similar easterly elephant named Viruupaaksha, which is bearing the eastern side of earth's surface on its head. Oh, Rama, the legatee of Raghu, that great easterly elephant Viruupaaksha is bearing the entire eastern earth along with its mountains and forests on its head. They circumambulated the great elephant in supplication, oh Rama, and on worshipping that elephant which is the protector of easterly direction they proceeded to netherworld duly splitting the earth. After splitting the eastern direction then they split opened the southern direction, and even in southern direction they beheld an elephant that is similar to a very great mountain, and that is sustaining southerly side of earth on its head, and on seeing that mammoth elephant of venerable character, namely Mahapadma, they went into an inordinate astonishment. On circumabulating that elephant Mahaapada in supplication, they the sixty thousand sons of great-souled Sagara scooped out the westerly direction. Even in the westerly direction those great mighty sons of Sagara beheld an infinite and mountain similar elephant of western direction, called Sumanasa. Circumabulating that elephant namely Sumanasa in western side, and even on asking about its well-being, then they reached the northern direction duly digging the earth. In the northern direction, oh, Rama, they beheld Bhadra, a snow-white elephant with an auspicious body bearing northern side of this earth. On touching that elephant reverently and also circumambulating it, those sixty thousands sons of Sagara further burrowed to the surface of the earth. The sons of Sagara have then gone to the auspicious northeast direction, iishaana digbhaaga, Shiva sthaana, and all those sons of Sagara have tunnelled the earth rancorously. The eyes of the princes are agitated in fury in construing the sage Kapila as the stealer of ritual-horse, and they furiously dashed towards him wielding crowbars, ploughs, and various trees and boulders, and shouting at him, 'stay, stay.'
Kapila burnt the sons of Sagara to ashes, which necessitated the famous story of the descent of the Ganga. But what's relevant for our purposes is that the Earth is described as being supported by four World Elephants: Virupaksha who supports the Eastern quarter, Mahapadma who supports the Southern quarter, Sumanasa who supports the Western quarter, and Bhadra who supports the North quarter. Also, the next chapter of the Bala Kanda of the Ramayana describes these four elephants as eloquent and revered by the gods and other beings:
And he that resplendent Amshuman beheld one of the four directional elephant of the earth which is being venerated by gods, monsters, demons, imps, vultures and serpents.... All the elephants that safeguard the directions of earth which have the faculties of articulation and eloquence have adored Amshuman and motivated him by saying, 'you will be going from here taking the horse.'
In any case, the four World Elephants are also described in this chapter of the Srimad Bhagavatam, albeit with different names:
On the top of Lokāloka Mountain are the four gaja-patis, the best of elephants, which were established in the four directions by Lord Brahmā, the supreme spiritual master of the entire universe. The names of those elephants are Ṛṣabha, Puṣkaracūḍa, Vāmana and Aparājita. They are responsible for maintaining the planetary systems of the universe.
World Tortoise: The 8th century Hindu astronomer Lalla mentions it as one of the Puranic theories he is trying to refute in this chapter of his Shishyavriddhida Tantra:
Some say that the earth is supported by a tortoise, a serpent, a boar, an elephant, or by mountain ranges.... If the earth is supported by tortoise and other things, by whom are they supported in space? If these can reman in space (unsupported), what prevents the earth from remaining thus (i.e. unsupported)?
And it's described in this chapter of the Devi Bhagavatam:
By Whose command the wind supports the water; the water supports the tortoise; the tortoise supports the Ananta and the Ananta supports the earth; the earth supports the oceans, mountain and all the jewels.
But I don't know anywhere else in Hindu scripture where this notion is mentioned.
World Serpent: This is the most common theory found in Hindu scripture. Here's how this chapter of the Vishnu Purana describes it:
The extent of the surface of the earth has been thus described to you, Maitreya. Its depth below the surface is said to be seventy thousand Yojanas, each of the seven regions of Pátála extending downwards ten thousand. These seven, worthy Muni, are called Atala, Vitala, Nitala, Gabhastimat, Mahátala, Sutala, and Pátála.... Below the seven Pátálas is the form of Vishńu, proceeding from the quality of darkness, which is called Śesha, the excellencies of which neither Daityas nor Dánavas can fully enumerate.... Śesha bears the entire world, like a diadem, upon his head, and he is the foundation on which the seven Pátálas rest. His power, his glory, his form, his nature, cannot be described, cannot he comprehended by the gods themselves. Who shall recount his might, who wears this whole earth, like a garland of flowers, tinged of a purple dye by the radiance of the jewels of his crests.
This chapter of the Adi Parva of the Mahabharata describes how Adiseshan became the supporter of the Earth. It was a duty given to him by Brahma as a result of his Tapasya:
Then Sesha replied, 'O divine Grandsire, this is the boon desired by me; viz., may my heart always delight in virtue and in blessed ascetic penances, O Lord of all!' Brahman said, 'O Sesha, I am exceedingly gratified with this thy self-denial and love of peace. But, at my command, let this act be done by thee for the good of my creatures. Bear thou, O Sesha, properly and well this Earth so unsteady with her mountains and forests, her seas and towns and retreats, so that she may be steady.' Sesha said, 'O divine Lord of all creatures, O bestower of boons, O lord of the Earth, lord of every created thing, lord of the universe, I will, even as thou sayest hold the Earth steady. Therefore, O lord of all creatures, place her on my head.' Brahman said, 'O best of snakes, go underneath the Earth. She will herself give thee a crevice to pass through. And, O Sesha, by holding the Earth, thou shalt certainly do what is prized by me very greatly.' ... Then the elder brother of the king of the snakes, entering a hole, passed to the other side of the Earth, and holding her, supported with his head that goddess with her belt of seas passing all round.' Brahman said, 'O Sesha, O best of snakes, thou art the god Dharma, because alone, with thy huge body, thou supportest the Earth with everything on her, even as I myself, or Valavit (Indra), can.' ... The snake, Sesha, the lord Ananta, of great prowess, lives underneath the Earth, alone supporting the world at the command of Brahman. And the illustrious Grandsire, the best of the immortals, then gave unto Ananta the bird of fair feathers, viz., the son of Vinata, for Ananta's help.'"
I'm not sure how Vishnu's bird Garuda helps Vishnu's serpent Adiseshan support the Earth though. In any case, here's how this chapter of the Srimad Bhaagavatam describes Adiseshan supporting the Earth:
My dear King, approximately 240,000 miles beneath the planet Pātāla lives another incarnation of the Supreme Personality of Godhead. He is the expansion of Lord Viṣṇu known as Lord Ananta or Lord Saṅkarṣaṇa.... This great universe, situated on one of Lord Anantadeva’s thousands of hoods, appears just like a white mustard seed. It is infinitesimal compared to the hood of Lord Ananta.
So as you can see, there's many different world animal theories, and sometimes these animals are described as stacked on top of each other. In any case, some people (like ISKCON members for instance) reconcile these theories with our understanding of the world by saying that the Earth is just Jambudvipa or even just Bharatavarsha, and that it is some much larger entity, like the entire physical Universe, which is the Bhuloka carried by Adiseshan.
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