We study the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as $\varepsilon$, the distance between insulators, approaches to 0. In 2D, we prove an upper bound of the gradient to be of order $\varepsilon^{-α}$, where $α= 1/2$ when $p \in(1,3]$ and any $α> 1/(p-1)$ when $p > 3$. We provide examples to show that this exponent is almost optimal. In dimensions $n \ge 3$, we prove an upper bound of order $\varepsilon^{-1/2 + β}$ for some $β> 0$, and show that $β\nearrow 1/2$ as $n \to \infty$.

39 pages. Theorem 1.3 is extended to all dimensions