Let $K$ be a function on a unimodular locally compact group $G$, and denote by ${{K}_{n}}\,=\,K\,*\,K\,*\cdots *\,K$ the $n$-th convolution power of $K$. Assuming that $K$ satisfies certain operator estimates in ${{L}^{2}}\left( G \right)$, we give estimates of the norms ${{\left\| {{K}_{n}} \right\|}_{2}}$ and ${{\left\| {{K}_{n}} \right\|}_{\infty }}$ for large $n$. In contrast to previous methods for estimating ${{\left\| {{K}_{n}} \right\|}_{\infty }}$, we do not need to assume that the function $K$ is a probability density or nonnegative. Our results also adapt for continuous time semigroups on $G$. Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.