For $ξ\geq 0$, Liouville first passage percolation (LFPP) is the random metric on $\varepsilon \mathbb Z^2$ obtained by weighting each vertex by $\varepsilon e^{ξh_\varepsilon(z)}$, where $h_\varepsilon(z)$ is the average of the whole-plane Gaussian free field $h$ over the circle $\partial B_\varepsilon(z)$. Ding and Gwynne (2018) showed that for $γ\in (0,2)$, LFPP with parameter $ξ= γ/d_γ$ is related to $γ$-Liouville quantum gravity (LQG), where $d_γ$ is the $γ$-LQG dimension exponent. For $ξ> 2/d_2$, LFPP is instead expected to be related to LQG with central charge greater than 1. We prove several estimates for LFPP distances for general $ξ\geq 0$. For $ξ\leq 2/d_2$, this leads to new bounds for $d_γ$ which improve on the best previously known upper (resp.\ lower) bounds for $d_γ$ in the case when $γ> \sqrt{8/3}$ (resp.\ $γ\in (0.4981, \sqrt{8/3})$). These bounds are consistent with the Watabiki (1993) prediction for $d_γ$. However, for $ξ> 1/\sqrt 3$ (or equivalently for LQG with central charge larger than 17) our bounds are inconsistent with the analytic continuation of Watabiki's prediction to the $ξ>2/d_2$ regime. We also obtain an upper bound for the Euclidean dimension of LFPP geodesics.