How is it possible for current to flow so fast when charge flows so slowly?
We know electrons travel very slowly while charge travels at ~the speed of light.
Answer
What is being confused here is not the flow of "current" but rather the transmission of energy.
The individual electrons in a wire move very slowly, as they can be modeled as constantly colliding with atoms (yes, this is a naive classical model, no quantum) and bouncing around randomly in the manner of a gas (the term "electron gas" is real and not inappropriate at all). Electric current is the very slow flow of this electron gas through the wire when an electric field is present. The term "flow of current" actually is misleading - there is no such "substance" called "current", current is a flow. "Flow of charges" or more specifically (in this case - in others, it may differ!) "flow of electrons" makes more sense. (After all, we don't talk about "current" as a substance which is contained within a river and which is what does the "flowing", i.e. "flow of current in the river", rather we talk of flowing "water in" the river, and "the current" means the flow of water.) See:
http://amasci.com/miscon/eleca.html#cflow
Energy, however, is not transmitted by one electron moving all the way around the circuit to the load, but rather through waves in the electrons and more importantly, the associated electric field. It's the same way that mechanical energy is transmitted in, say, a pole that is pushed from one end. The pole compresses slightly, and a sound wave thus appears, initially containing all the energy within your "push", and then travels down it, progressively distributing that energy amongst all the atoms within the pole until they are all moving in a single direction (here I imagine the pole pushed in a vacuum, as in interstellar space, with no other forces acting). The same goes with electrons in the circuit - though I should point out the following model is a bit simplistic but is more to convey the point of how the energy is transmitted than to detail the actual behavior of the electrons, which involves quantum mechanics and is subject to many of the same caveats as one sees within in an individual atom or molecule. But in this loose sense, when you throw the switch, now an electromagnetic wave travels down, setting the electrons ahead in motion and thus distributing its energy throughout the circuit. Of course, the core atoms of the metal are relatively fixed despite the electron motion, so the latter will tend to lose that energy to collision with them, unlike the pole where everyone, atoms and electrons together, start going in synchrony, and thus you have to keep supplying energy to them with a power source like a battery or generator which effectively keeps "pushing the pole" and thus keeps energy going into it - now think about a pole that is now not in vacuum but in molasses, and you have to keep pushing it to keep it moving. This pushing on atoms, of course, is how electrical devices can use electrically transmitted energy to do useful tasks.
Electromagnetic waves, and sound waves, thus energy, travel much faster than the electrons and the atoms in both the circuit and pushed pole. Energy is what lights up your light bulbs, and energy is what makes your computer operate. Since energy travels fast, these devices start operating "at the flick of a switch".
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