Science doesn't deny the existence of the supernatural.
Science is merely (very roughly speaking) the process of investigating things that are consistently and repeatedly observable (or which has or had effects that are so observable).
This doesn't entail denying the existence of the supernatural or things without direct or indirect observable effects. There are a wide range of beliefs about the supernatural among scientists.
The issue is just that you can't investigate things through observation, if those things aren't directly or indirectly observable, so science doesn't say anything about those things - that's roughly what methodological naturalism is. Those things are instead in the realm of philosophy and epistemology (it's arguable whether one can have justifiable beliefs in unobservable things... or undeducible things).
In this sense, someone who wants to say that science fundamentally can't investigate the deity they believe in would need to commit themselves to the position that said deity never directly acted in reality (e.g. deism), for otherwise there would be effects we could observe that science could investigate.
* Every time the natural/supernatural distinction comes up, I feel I should point out that it's typically believers in the supernatural who are insisting on that distinction. I reject the supernatural for a similar reason to why I reject srofmogruesh: I don't know what that is, and none of the things that seem to exist have that label. Attempts to define the supernatural have severe issues. When people say that others presuppose that the "supernatural" doesn't exist, that generally comes across as them trying to distract from the fact that they can't meet their burden of proof for their supernatural claims.
Comparison to a meteor impact
If we consider one particular (natural) meteor impact as an example:
This is a one-time event, so it's not consistently and repeatedly observable directly. We may not have direct observation of it at all.
But we can consistently and repeatedly observe the soil around the impact event, and we can take measurements and compare that to soil elsewhere, and we can simulate the effects of such an impact on the surrounding area, and take measurements to look for those effects.
Through these means, we can gain confidence that a meteor impact indeed happened.
Replace "meteor impact" with some deity flooding the world or whatever, and it could be investigated similarly. Note that none of what I said above relates to where the meteor originally came from (even if there could be evidence for that too), so if someone says a deity summoned a rock and slammed it into the ground, all these lines of evidence would be the same. This raises the question of how you can differentiate "it came from space" from "God did it" - more on that later.
For a meteor impact, a strong line of evidence is the comparison to other meteor impacts (and that also relates to how we know the effects of meteor strikes). For a "miracle", this may explicitly be a unique event, so it may be harder or less viable to compare it to similar things. But there are often still similarities - if we're trying to investigate a worldwide flood, local floods would be somewhat similar, we can observe soil layering elsewhere, we can observe the effects water has on things elsewhere, and so on.
Getting to the supernatural
Science can investigate whether there was a meteor impact.
If someone says their deity caused a meteor impact, we can investigate whether the effect happened, but there's still a gap between the cause and the effect that's difficult to cross. It's hard to end up at "God did it" instead of there having been some natural cause, especially if the existence of the proposed deity would entail a whole bunch of other things that are contrary to what we see, e.g. the problem of evil/suffering or divine hiddenness. See also: God of the gaps.
I wouldn't say this effect / supernatural cause boundary is impossible to cross:
- If some text were to spontaneously materialise in the air all across the world at the exact same time, in the language of the locals, that would go a long way to making a case for the existence of some intelligent ultra-powerful being. As Clark said, "sufficiently advanced technology is indistinguishable from magic", and this is still a long way from the truth of any given religion, but we'd certainly be a lot closer than where we are now.
- Claims of levitation, telekinesis, mind reading, etc. should be exceedingly easy to scientifically verify, if those are things someone can reliably do in ways that can't easily be explained through more mundane ways (e.g. hot reading and cold reading). See also: James Randi's million dollar paranormal challenge.
The issue with supernatural claims is that they often don't cross the cause/effect boundary to justify why that particular explanation is the best one, they avoid specificity, they avoid claims that can be verified (a claim that can be verified can also be falsified), they mutate to avoid falsification, etc.
Let's say someone says e.g. God prevented a car accident - maybe they saw a deer on the side of the road which caused them to slow down. If we had video evidence of the deer suddenly materialising out of nothing, that might still be a few steps from "God did it", but we can certainly say that would be outside of the laws of physics as we understand them. But we don't have reliable evidence of such things. More likely, we may have evidence of the deer walking or running up to that spot, with theists saying God made the deer do that. But if God acts through natural processes, that's largely indistinguishable from God not intervening at all (unless there's unambiguously a message there - the spontaneous materialising of clear and complex text messages as above may be attributed to a powerful being even if it happens due to natural processes).
* Some theists say that DNA contains messages (or "information" or "instructions"), but that's only the case if you use that term very loosely, or if you make some rather big leaps to get to a deity actually wanting us to "read" those things (which we couldn't do for most of human history). DNA is just chemicals that interacts with other chemicals in certain ways to achieve certain results (disclaimer: I'm not a biologist), and evolution explains how we ended up with the DNA we ended up with. This can't reasonably be considered a message intended to be read by humans.
History?
I didn't mention history, but a lot of what I said above about science could apply to history as well. You have some observations like historical writing, ancient ruins, etc., and you're trying to come up with the best explanation for those observations. You can't say much for things you don't have observations for.