Something is supernatural if it defies the natural laws of physics. Now you might be thinking, "isn't it natural for the supernatural to defy the laws of physics? Wouldn't that make the supernatural, natural?" The answer is YES, the supernatural is natural. But it's a different kind of natural. You're confused because the definitions aren't clear. I will try my best to make them clear..
Let's imagine the word "NATURAL" written on a white board in all capital letters. Let's use the word "NATURAL" to describe anything that exists. For example, watermelons are "NATURAL" because watermelons exist. If a god existed, he would be "NATURAL" as well. For the sake of argument, let's say a god existed.
Now let's draw two lines branching from the word "NATURAL" with one line leading to the word "natural" (lower cases) and the second line leading to the word "supernatural." What's the difference between "natural" and "supernatural," you ask? Well, things that are "natural" can be predicted if all the variables are known. Watermelons would be "natural" because watermelons follow consistent laws that are predictable. That is, science can predict how a watermelon will grow in an environment given known temperature, weather conditions, climate, etc. On the other hand, things that are "supernatural" aren't restricted by "natural" laws and cannot be predicted by science. A god would be "supernatural" because a god doesn't have to abide by "natural" laws. A god can create a watermelon out of thin air without actually having to grow one. It doesn't matter if it's too hot or too cold or if there's a severe drought. A god can miraculously create a watermelon regardless of the "natural" conditions normally required to grow a watermelon.
So there you have it, there are two kinds of NATURAL (two kinds of existences). These two kinds of NATURAL are called "natural" and "supernatural". From these definitions, you can see that both the watermelon and the god are "NATURAL" because they both exist. However, only the watermelon is "natural" because the watermelon is bound by consistent laws that can be predicted by science. The god is also "NATURAL" because he exists, but he's not "natural" because he doesn't have to follow consistent laws that would allow science to make predictions. The god is "NATURALLY supernatural" as opposed to "NATURALLY natural."
I just want to add that science can never prove anything "supernatural" because science is built around the assumption that all phenomena can be explained by "natural" laws. Even if "supernatural" things do exist, science will never be able to prove them because science is, by definition, the study of the "natural" world. Trying to use science to prove the supernatural is like trying to use a compass to tell time.