Many triangulated odd-spheres

It is known that the $(2k-1)$-sphere has at most $2^{O(n^k \log n)}$ combinatorially distinct triangulations with $n$ vertices, for every $k\ge 2$. Here we construct at least $2^{\Omega(n^k)}$ such triangulations, improving on the previous constructions which gave $2^{\Omega(n^{k-1})}$ in the general case (Kalai) and $2^{\Omega(n^{5/4})}$ for $k=2$ (Pfeifle-Ziegler). We also construct $2^{\Omega\left(n^{k-1+\frac{1}{k}}\right)}$ geodesic (a.k.a. star-convex) $n$-vertex triangualtions of the $(2k-1)$-sphere. As a step for this (in the case $k=2$) we construct $n$-vertex $4$-polytopes containing $\Omega(n^{3/2})$ facets that are not simplices, or with $\Omega(n^{3/2})$ edges of degree three.