I'm very new to quantum computing. I was just wondering if a quantum computer simulator could be faster than a normal computer when running on a normal computer. Could it?
I could argue your answer in either way, based on semantics.
The basic answer is as Michele said: no. The classical simulation of a quantum algorithm is itself a classical algorithm. So that cannot outperform the best possible classical algorithm.
However, you can turn that around a little bit. It is possible that the classical simulation of a quantum algorithm can outperform the best classical algorithm currently known. You get a new "quantum inspired" algorithm that performs better than anything we had before. In that sense, it can be a valuable research direction, for example.
No, it could not.
First of all, when you simulate a quantum computer, you simulate it at executing a quantum algorithm on a specific problem instance. If you manage to simulate such an execution on a classical computer, this means that the result of your quantum computation could be achieved by means of a classical algorithm, so it is not interesting.