We know that planar graphs have $O(1)$ degree.

We know balanced (each color has same number of vertices) complete bipartite graphs have genus $O(n^2)$.

If maximum and average degree are $O(n^\alpha)$ where $\alpha\in[0,1]$ then is genus also $O(n^\alpha)$?

If maximum degree is $O(n^\alpha)$ where $\alpha\in[0,1]$ then is genus also $O(n^\alpha)$? (this is weaker than $1.$)