  We consider the hypergraph Tur\'an problem of determining $\mathrm{ex}(n,
S^d)$, the maximum number of facets in a $d$-dimensional simplicial complex on
$n$ vertices that does not contain a simplicial $d$-sphere (a homeomorph of
$S^d$) as a subcomplex. We show that if there is an affirmative answer to a
question of Gromov about sphere enumeration in high dimensions, then
$\mathrm{ex}(n, S^d) \geq \Omega(n^{d + 1 - (d + 1)/(2^{d + 1} - 2)})$.
Furthermore, this lower bound holds unconditionally for 2-LC spheres, which
includes all shellable spheres and therefore all polytopes. We also prove an
upper bound on $\mathrm{ex}(n, S^d)$ of $O(n^{d + 1 - 1/2^{d - 1}})$ using a
simple induction argument. We conjecture that the upper bound can be improved
to match the conditional lower bound.
