We prove 'polynomial in k' bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree n and weight k. When n = 1, 2 our bounds agree with the conjectural bounds on the aforementioned size, while the lower bounds match for all n ≥ 1. For an L 2 -normalised Siegel cusp form F of degree 2, our bound for its supnorm is O ǫ (k 9/4+ǫ ). Further, we show that in any compact set Ω (which does not depend on k) contained in the Siegel fundamental domain of Sp(2, Z) on the Siegel upper half space, the sup-norm of F is O Ω (k 3/2−η ) for some η > 0, going beyond the 'generic' bound in this setting.