An abstract simplicial complex $\mathbf{F}$ is a non-uniform hypergraph without isolated vertices, whose edge set is closed under taking subsets. The extremal number $\mathrm{ex}(n,\mathbf{F})$ is the maximum number of edges in an $\mathbf{F}$-free simplicial complex on $n$ vertices. This extremal number is naturally related to the generalised Turán numbers of certain underlying hypergraphs. Making progress in a problem raised by Conlon, Piga, and Schülke, we find large classes of simplicial complexes whose extremal numbers are determined by the respective generalised hypergraph Turán numbers. We also provide simplicial complexes for which such a relation does not hold.