Quick edit: I should point out that, at least in theory, the accused witches being immersed (usually by being thrown in water, or being thrown in water with a rope tied around them, or being thrown in water tied to a chair with or without a rope tied to it) were not supposed to drown if they went under. They were supposed to be hauled out and set free.

Yes, the practice of "swimming" witches was used, and yes, even people at the time who believed in witches recognized that this is a very dubious means of testing. In England, the practice seems to have become common in the seventeenth century. According to a footnote in Alan Macfarlane's Witchcraft in Tudor and Stuart England,

Scot, Discovery, p. 255, implied that the water ordeal was not used in England in 1584, while Perkins in 1608 (Damned Art, p. 206) only referred to its use ‘in other countries’. Yet it seems to have been in fairly widespread use by 1616, according to John Cotta, Triall of Witchcraft (1616)

The most common explanation for the practice is given by King James VI and I in his very influential Daemonologie (1597), in which he describes that

it appears that God hath appoynted (for a super-naturall signe of the monstruous impietie of the Witches) the water shal refuse to receive them in his bosom, that have shaken off them the sacred Water of Baptisme, and wilfullie refused the benefite thereof.

He then goes on to say that witches are for this reason unable to cry, even though women are "able other-waies to shed teares at every light occasion when they will, yeah, although it were dissemblingly like the Crocodiles."

In 1612, an anonymous pamphlet appeared that described the methods of four magistrates conducting witch trials. One, Sir Gilbert Pickering of Titchmarsh, uses the method described, and the writer of the pamphlet defends the practice for the same reason James gave.

In 1625, the physician John Cotta, who wrote several books about magic and witchcraft, includes the practice in his book "The infallible True and Assured Witch" (the title alone stresses how important he finds it that only actual witches are punished). After describing how carefully and methodically witch trials should be conducted, he condemns the practice of "swimming" witches in particular, writing,

How farre our vulgar tryall of Witches, by the supposed miraculous indication and detection of them by the water, is different from this care of respect, this equitie, religion, or humanitie, common practise doth openly declare, when without allowance of any law, or respect of common civilitie, every private, rash, and turbulent person, upon his own surmise of a Witch, dare barbarously undertake by uncivill force and lawlesse violence, to cast poore people bound into the water, and there deteine them, for their own vaine and foolish lusts, without sense, or care of the shamefull wrong, or injury, which may befall oft-times innocents thereby.

Though this kinde of tryall of a Witch, might haply prove in it selfeworthy to be allowed, yet is it not in every private person justifiable, or tolerable, or without warrant of authorotie in any sort excusable. The manner therefore of this vulgar tryall, must needs with just and honest mindes, uncontroversedly, and undoubtedly, be rusticall, barbarous, and rude. (134, I have adjusted some letters to match modern usage, ex. v<->u)

He goes on to discuss at length whether we can credit that a witch could be cast out of water as a miracle, and ultimately rejects it. It is interesting that he does not deny that a woman might be rejected by the water for supernatural reasons, but rather questions who causes this to happen, God or the devil. He says that in general using such "signs" to prove a witch is judging them as "miracles of the Divell" and therefore these "supposed miraculous detection of Witches by the water" should be stripped "of any hopefull opinion that they can be of God" (137).

The practice of "swimming" witches was practiced by the infamous Witchfinder General, Matthew Hopkins, who was warned against it and forced to stop in 1645. Hopkins writes about the practice in his manual The Discovery of Witches from 1647. The book is composed as an imagined dialogue between someone questioning him and his answers. The question he uses pretty succinctly expresses the objections to it:

But there hath been an abominable, inhumane, and unmercifull tryall of these poore creatures, by tying them, and heaving them into the water; a tryall not allowable by Law or conscience, and I would faine know the reasons for that.

Hopkins defends the practice, though, saying that witches came to him "of their own accord" to have the test performed, because the devil whispered in their ear that he would help them pass the test, which they then go on to fail, proving how treacherous the devil is. King James's book is one of his favorite sources, and he cites James's explanation about baptism here. He also points out that witches can cry when accused of other crimes, but will not cry when accused of witchcraft.

Macfarlane writes that in the eighteenth and nineteenth century "Old women were still informally tried by being immersed in water" (88), but this was not the widespread or regulated practice of previous centuries.

Further reading: Cabell, Craig. 2006. Witch-Finder General: The Biography of Matthew Hopkins.
Elmer, Peter. 2016. Witchcraft Witch-Hunting, and Politics in Early Modern England.
Macfarlane, Alan. 1970, second edition 1999. Witchcraft in Tudor and Stuart England.

Here is a link to a thread with some info I wrote on the history and practice of "trials by water" during the Medieval period, namely that the purpose was not to kill the accused, and that during the period of its accepted use as a legal practice it was not specifically related to witchcraft accusations.