For an element to be conductive, it has to be metallic by nature. Salt (Sodium chloride) contains Sodium, which is a metal and therefore conductive. Chlorine, by itself, is not a metal and therefore not conductive.

Although chlorine + water -> HOCl + HCl
both HOCl & HCl disocciate into ions in water, making the water conductive.
Nevermind, why am i getting into this

HCl dissolved in water can conduct electricity. It has to do with fully dissociating in water and the ions present in solution. It doesn't have to be metallic (even though Na+ is not the same as Na anyway).

You are right, I'm sorry. It is an old kicker from my elementary school that a metal has to be present.

To clarify: NaCl conducts electricity because, when dissolved in water, breaks down into ions and therefore a matter of charged particles that electricity can travel across. The same thing happens with HCl.
Chlorine by itself cannot, however, be broken up into ions as far as I'm concerned.
Also, isn't Na+ just the charged ion of Na?

Yes, Na+ is just the cation of Na, but it has completely different properties and reacts differently than Na would.

And you clearly will die if you drop a toaster that's plugged in into a bath. Try it for yourself. Pure water does not conduct electricity, but water is never pure. Pure water is a theoretical concept, really.

I have made this exact experiment on myself (with tap water) with a toaster and a shaking finger. Gladly, nothing came out of it.

Sure, you will not have pure water in any other place than a laboratory. But how unclean do you think tap water is? It contains some calcium, yes, and some traces of some other elements but it is no where near anything that's potentially dangerous in an electricity conductive sense.

You can just google it, honestly. Putting an electric appliance in a bathtub of water can be fatal. And I think mythbusters did an episode on it with conclusions that support this.

I'm dangerous!