We give upper bounds for $L^p$ norms of eigenfunctions of the Laplacian on compact hyperbolic surfaces in terms of a parameter depending on the growth rate of the number of short geodesic loops passing through a point. When the genus $g \to +\infty$, we show that random hyperbolic surfaces $X$ with respect to the Weil-Petersson volume have with high probability at most one such loop of length less than $c \log g$ for small enough $c > 0$. This allows us to deduce that the $L^p$ norms of $L^2$ normalised eigenfunctions on $X$ are a $O(1/\sqrt{\log g})$ with high probability in the large genus limit for any $p > 2 + \varepsilon$ for $\varepsilon > 0$ depending on the spectral gap $λ_1(X)$ of $X$, with an implied constant depending on the eigenvalue and the injectivity radius.