Suppose $G$ is a finite simple group of order $n$ with a nontrivial representation of degree $d$. Then $G$ is isomorphic to a subgroup of $U(d)$. By Collins's sharp version of Jordan's theorem (https://www.degruyter.com/document/doi/10.1515/JGT.2007.032/html), $G$ has an abelian normal subgroup of index at most $(d+1)!$, which must be trivial since $G$ is simple, so $|G| \leq (d+1)!$. Rearranging, $d \gtrsim \log n / \log\log n$.

Collins's work builds on work of Weisfeiler that I think was unfinished by the time of his disappearance.

Suppose $G$ is a finite simple group of order $n$ with a nontrivial representation of degree $d$. Then $G$ is isomorphic to a subgroup of $U(d)$. By Collins's sharp version of Jordan's theorem (https://www.degruyter.com/document/doi/10.1515/JGT.2007.032/html), $G$ has an abelian normal subgroup of index at most $(d+1)!$, which must be trivial since $G$ is simple, so $|G| \leq (d+1)!$. Rearranging, $d \gtrsim \log n / \log\log n$.

Collins's work builds on work of Weisfeiler that I think was unfinished by the time of his disappearance.

Edit: Specializing to simple $G$ actually reduces Collins's paper to a reference to a reference to the paper of Seitz and Zalesskii (https://www.sciencedirect.com/science/article/pii/S0021869383711324?via%3Dihub) mentioned by David Craven in the comments, so that's really the heart of the matter. We thereby get the slightly stronger bound $n \leq (d+1)!/2$ (for sufficiently large $d$). Apart from alternating groups I think you can read out a much stronger bound like $n \leq \exp O((\log d)^2)$, or $d \geq \exp \Omega( (\log n)^{1/2})$ (I am guessing the next worst case is $\mathrm{SL}_n(2)$).