During the electrolysis of $\ce{H2SO4}$, $\ce {OH-}$ ions are more reactive than $\ce {SO4^{2-} }$ ions and so they dissociate. as $\ce {OH-}$ ions dissociate, they are the negative ions that carry current. So when the concentration of $\ce {H2SO4}$ is increased, the concentration of $\ce {OH-}$ ions actually reduces, but then the current still increases.
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Near room temperature, considering solutions of water and sulphuric acid, the maximum conductivity is at about 30% sulfuric acid. For pure water or pure sulfuric acid the conductivity is orders of magnitude lower.
In pure water, there are only $10^{-7}$M of $\ce{H+}$ and $\ce{OH-}$ to carry current.
When a relatively small amount of sulfuric acid is added, it dissociates fully to $\ce{H+}$ and $\ce{HSO4-}$. For example if you increase the concentration of sulfuric acid from 0M to 0.01 M, the number of ions in the solution increase by a factor of $10^5$.
