  High-quality single crystals of CoTiO$_3$ are grown and used to elucidate in
detail structural and magnetostructural effects by means of high-resolution
capacitance dilatometry studies in fields up to 15 T which are complemented by
specific heat and magnetization measurements. In addition, we refine the
single-crystal structure of the ilmenite ($R\bar{3}$) phase. At the
antiferromagnetic ordering temperature $T_\mathrm{N}$, pronounced
$\lambda$-shaped anomaly in the thermal expansion coefficients signals
shrinking of both the $c$ and $b$ axes, indicating strong magnetoelastic
coupling with uniaxial pressure along $c$ yielding six times larger effect on
$T_\mathrm{N}$ than the pressure applied in-plane. The hydrostatic pressure
dependency derived by means of Gr\"uneisen analysis amounts to $\partial
T_\mathrm{N}/ \partial p\approx 2.7(4)$~K/GPa. The high-field magnetization
studies in static and pulsed magnetic fields up to 60~T along with high-field
thermal expansion measurements facilitate in constructing the complete
anisotropic magnetic phase diagram of CoTiO$_3$. While the results confirm the
presence of significant magnetodielectric coupling, our data show that
magnetism drives the observed structural, dielectric, and magnetic changes both
in the short-range ordered regime well-above $T_\mathrm{N}$ as well as in the
long-range magnetically ordered phase.
