Is there a one-word term for "grows by square root"?

The asymptotic growth of a root is known as "sublinear growth". Another term is "fractional-power" growth but it sounds odd imo.

The sublinear term originates from linear algerba. It can be used not only for square-root but for any $n-th$ root complexity.

In strict terms as sublinear function is a function that satisfies the following properties:

• Positive Homogeneity: $f(kx) = k^n f(x) \quad n\in\mathbb{R^+_0}$ and
• Subadditivity: $f(x+y) \leq f(x) + f(y)$

Roots satisfy both. The proof of positive homogeneity is trivial. As far as the proof of subadditivity is concerned you can check this post

Other seemingly sublinear functions (such as log) do not satisfy these properties therefore, in a sense you can separate them from sublinears.