In this work we obtain rates of convergence for two moderately interacting stochastic particle systems with singular kernels associated to the viscous Burgers and Keller-Segel equations. The main novelty of this work is to consider a non-locally integrable kernel. Namely for the viscous Burgers equation in $\mathbb{R}$, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some Bessel space with a rate of convergence of order $N^{-1/8}$, on any time interval. The same holds for the genuine empirical measure in Wasserstein distance. In the case of the Keller-Segel equation on a $d$-dimensional torus, we obtain almost sure convergence of the mollified empirical measure to the solution of the PDE in some $L^q$ space with a rate of order $N^{-\frac{1}{2d+1}}$. The result holds up to the maximal existence time of the PDE, for any value of the chemo-attractant sensitivity $\chi$.