It's difficult to imagine how the idea of free-will could be "demonstrated" to exist because of poorly it is often defined or perhaps how incoherent the idea is.
There are two definitions commonly used for free will.
The first is used by proponents of libertarian free-will (it would seem) that free-will is something kind of 'supernatural' in that it isn't so much subject to causality in the same way that all other parts of the natural world are. You could definitely find proponents of libertarianism that don't use this definition though, I have to speak in general terms.
The second is a kind of political-social free will. The idea that we're free to act as we desire.
The problem with free will is that what we do is ruled by our desires. The principle stands that you always act in the way you desire to act. More or less by definition. And findings in neuroscience and psychology suggest that our desires are moulded by a combination of genetics and our environment.
That's not a problem for the second 'concept' of free will as it involves being able to act in accordance with our desires and makes no demands about where our desires must come from.
This IS a problem for the first kind since, as Schopenhauer said, you cannot will what you will. That is, you cannot choose what you desire even on a logical level. If you choose what you desire, do you choose to desire what you desire? And then to desire what you desire what you desire? etc...
Similarly, there's it's not even that obvious that our subjective experience is even trying to fool us that we have free will. Thoughts about what we want or how we should act in the present moment arise as if from nowhere. When you react to a stimulus and feel as though you have a set of behaviours to choose from, in no way to you choose which behaviours your sub-conscious will bring up first. Nor do you really, when you focus enough, choose the final outcome. This is an argument often made by Sam Harris (although he's a bit of a poppy philosopher and perhaps isn't well respected in ultra-academic circles, I did find this argument extremely interesting when I first heard it and it can be found in this presentation: https://www.youtube.com/watch?v=JM3raA1EwrI)
I think your best bet is to watch that presentation. I found it hard to deny what he was saying about the nature of your subjective experience as it relates to whether or not it even FEELS like we have freewill. The comparison he sometimes makes is, when you're writing and you make a spelling mistake, in no way did your freewill have a part in doing that. But similarly, when you're writing and thinking up your next word, the emergence of the words you decide to keep in the writing is similarly mysterious and out of the blue. I can't deny that.