We study long-range Bernoulli percolation on $Zd$ in which each two vertices $x$ and $y$ are connected by an edge with probability $1-\exp (-\beta \parallel x-y\parallel -d-\alpha )$. It is a theorem of Noam Berger (CMP, 2002) that if then there is no infinite cluster at the critical parameter $\beta c$. We give a new, quantitative proof of this theorem establishing the power-law upper bound [ \mathbf{P}_{\beta_c}\bigl(|K|\geq n\bigr) \leq C n^{-(d-\alpha)/(2d+\alpha)} ] for every $n\ge 1$, where $K$ is the cluster of the origin. We believe that this is the first rigorous power-law upper bound for a Bernoulli percolation model that is neither planar nor expected to exhibit mean-field critical behaviour. As part of the proof, we establish a universal inequality implying that the maximum size of a cluster in percolation on any finite graph is of the same order as its mean with high probability. We apply this inequality to derive a new rigorous hyperscaling inequality $(2-\eta )(\delta +1)\le d(\delta -1)$ relating the cluster-volume exponent $\delta$ and two-point function exponent $\eta$.