How does strong acids conduct electricity in water?
In water at 25 ºC $[H+]=[OH-]=10^{-7}M$, so a concentration of $[H+]=10^{-6}M$ is very low. With a strong acid like HCl you have to dissolve only around 35 micrograms in a liter of water to get this concentration.
Sea water on the other hand has a high conductivity and, in average, 35 grams per liter of salts dissolved in it. If you dissolve 35 grams of HCl en 1 liter of water you have $[H+]=1M$ and surely you can expect a high conductivity.
These are gross numbers, but I think they help to get the idea.
Conductivity of a solution depends on the free ions present on the mixture. With a strong acid you can get a high concentration of ions using a little quantity of acid. With a weak acid you need much more.