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 $2^{\mathrm{\Omega }(n^k)}$ such triangulations, improving on the previous constructions which gave $2^{\mathrm{\Omega }(n^{k-1})}$ in the general case (Kalai) and $2^{\mathrm{\Omega }(n^{5\mathrm{/}4})}$ for $k=2$ (Pfeifle-Ziegler). We also construct $2^{\mathrm{\Omega }\left(n^{k-1+\frac{1}{k}}\right)}$ 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 $\mathrm{\Omega }(n^{3\mathrm{/}2})$ facets that are not simplices, or with $\mathrm{\Omega }(n^{3\mathrm{/}2})$ edges of degree three.