Once again, the point needs to be made that we don't have clear, agreed, definitions of the words "natural" and "supernatural". This is not a surprising fact. These are ordinary English words, used on a daily basis in informal discourse. We know that the words used in this way, in ordinary language, are typically quite fuzzy. Wittgenstein showed this with the example of the word "game", whose meaning is surprisingly difficult to nail down. It is very arguable - and Wittgenstein makes the argument - that the various things we call "games" have no one feature possessed by all. Rather, they bear a kind of family resemblance to each other. You can find paradigm examples that everyone calls a "game" - chess perhaps - but it might not have any feature in common with absolutely everything that we call a game. Still, most things that we call "games" do have features in common with many other things that we call "games". For most practical purposes, we can use the word "game" without too much misunderstanding.
Discussion of whether science can investigate "the supernatural" often seems to involve one party or another to the discussion assuming that "supernatural" has a clear meaning, when it simply doesn't. There are various phenomena that are commonly called "supernatural", but it's not really clear what they have in common.
Think of ghosts, gods, angels, demons, and astrological influences. You might think that what is involved with some of these is the claim that a disembodied mind exists, but gods are not always thought of as disembodied, and even ghosts are sometimes thought of as having a body of a certain kind - some sort of very thin stuff (the word "ectoplasm" is sometimes used here). In the past, spirits were not thought of as disembodied in the sense of being totally immaterial: they were made of "spirit" or "pneuma", a kind of super-thin material. The idea of a totally immaterial mind seems much more recent. Perhaps it existed before Descartes, but it's not clear where it appears in ancient thought or in mythology. And of course, astrological influences are not usually thought of as involving minds at all, although there is some kind of influence that we (perhaps) don't want to call "physical".
Perhaps some clear definition of "physicality" or "body" can be given, and people who call themselves "naturalists" can claim to be rejecting the "supernatural" when they reject the existence of minds that are totally without any physicality or embodiment. However, it would be unusual to find anyone who believes in the existence of such "supernatural" minds while claiming that they have no power to influence the physical world in ways that can be sensed. If a "supernatural" mind has the power to interact with the physical world, then we can study it. We can't study its physical substance, of course, if it has no physical substance, but we can study its powers, its behaviour, perhaps its motivations. Its non-physicality may limit the conclusions we can draw about it, but as long as enough is asserted about what it does we can certainly study whether there is evidence for or against the claim that it exists.
Thus, on one definition of "the supernatural", it is not beyond science to examine any claims at all about the supernatural.
But "the supernatural" might be defined in other ways. If we define "the supernatural" as "whatever cannot be studied by science", then of course it is true by definition that science cannot study the supernatural. However, this sort of definitional fiat tells us absolutely nothing about what sorts of things actually do exist and what sorts of things actually are open to investigation by science.
If we define "the natural" to mean "whatever exists" and "the supernatural" to mean "whatever is not natural", then we have made it true by definition that whatever exists is "natural" and that nothing "supernatural" exists. It follows that science cannot study "the supernatural" because there is no such thing to study. That, however, does not prove that gods, ghosts, demons, astrological influences, etc., don't exist. Nor does it show that they can't be studied by science. It only shows us that if ghosts (for example) do turn out to exist, then they must be classified as "natural".
In short, you can't determine what sorts of things do or do not exist, or what sorts of things can or cannot be studied by science, simply by definitional fiat. The definitions of words don't control what exists in reality.
Again, say you define "supernatural" as meaning "can't be studied by science". Using that definition, you can then demonstrate (a simple semantic entailment) that "the supernatural cannot be studied by science".
What you can't do is then pretend that you defined "the supernatural" to mean (for example) "ghosts and gods" and conclude "ghosts and gods cannot be studied by science". That's equivocation on the word "supernatural".
When we say that science deals with the "natural", while religion deals with the "supernatural" we need to define our terms (reasonably) clearly, then use them consistently. If we define the "natural" so that it means "everything" then it will turn out to be the case that science can deal with whatever turns out to exist, but that tells us nothing about what sorts of things actually do exist. It leaves open the question of whether gods and ghosts exist, for example. If they do, then it suggests that science can deal with them.
Whenever we debate what can be investigated by science, whether science and religion are compatible, whether the existence of entities typically referred to by religions (such as gods) can be studied scientifically, and whether those things exist at all, it would be helpful if we made a conscientious effort to use the various terms consistently and to take note of how others define their terms if they bother to do so. This is difficult to do - language is slippery, and a point comes where it is too demanding to cross every single "t" and dot every single "i". Nonetheless, it is not playing games. It is important that we do this to avoid trapping ourselves in prisons of words, drawing conclusions that are not justified by the facts. It would also help us avoid a lot of distracting emotional attacks that merely make the debate more difficult to keep under control.