In this paper, for given mass $m>0$, we focus on the existence and nonexistence of constrained minimizers of the energy functional

$$I(u):=\frac{a}{2}{\int }_{{\mathbb{R}}^3}{\left|\mathrm{\nabla }u\right|}^{2}dx+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^3}{\left|\mathrm{\nabla }u\right|}^{2}dx\right)}^2-{\int }_{{\mathbb{R}}^3}F(u)dx$$

on $S_m:=\left\{u\in H^1({\mathbb{R}}^3):\Vert u{\Vert }_2^2=m\right\},$where $a,b>0$ and $F$ satisfies the almost optimal mass subcritical growth assumptions. We also establish the relationship between the normalized ground state solutions and the ground state to the action functional $I(u)-\frac{\lambda }{2}\Vert u{\Vert }_2^2$. Our results extend, nontrivially, the ones in Shibata (Manuscripta Math. 143 (2014) 221–237) and Jeanjean and Lu (Calc. Var. 61 (2022) 214) to the Kirchhoff type equations, and generalize and sharply improve the ones in Ye (Math. Methods. Appl. Sci. 38 (2015) 2603–2679) and Chen et al. (Appl. Math. Optim. 84 (2021) 773–806).