Helmut Wielandt discussed an old question (Chap. 2, Section 15, which can be dated back to Camille Jordan):
Let $g\neq 1$ be a permutation in some finite primitive permutation group $G$ of degree $n$. The minimal degree $m$ is defined to be the least number of points that $g$ permutes. It is known that if $m>3$, then $n$ is bounded by $$\frac{m^2}{4}\log\frac{m}{2}+m\left(\log\frac{m}{2}+\frac{3}{2}\right).$$
Question: Do we know a better upper bound today (with CFSG and O'Nan-Scott etc.)?
You seem to be aware of the answer to your own question, since you give the reference to the paper of Guralnick and Magaard, which classifies groups of minimal degree $\leq n/2$. Therefore $n \leq 2m$ with explicit exceptions list in the G--M paper. See also the previous paper of Liebeck and Saxl, which is easier and does $m < n/3$.
Even without CFSG and O'Nan--Scott, we know a sharp bound today. It's a result of Babai that $m \geq c \sqrt{n}$. Babai's proof is purely combinatorial and extends to primitive coherent configurations (e.g., strongly regular graphs), and it's close to sharp for the groups $S_m \wr S_2 \leq S_{m^2}$ and $S_m \leq \mathrm{Sym} (\binom{m}{2})$. An exactly sharp bound follows from recent work of Sun and Wilmes (https://arxiv.org/abs/1510.02195), which implies a classification of primitive coherent configurations where the minimal degree is $\leq n^{2/3} (\log n)^{-C}$.
I know there is an accepted answer, but this might be helpful as well.
There is a very recent paper of Burness and Guralnick (see https://arxiv.org/abs/2112.03967), where the primitive groups with $m\ge 2n/3$ are classified (in Theorem 4). Of course, the results are CFSG-based.
The introduction of this paper contains very detailed history of bounding the minimal degrees and the fixed point ratios of finite primitive groups, which should be helpful.
