In this paper, we study the normalized solutions of the Schrödinger system with trapping potentials \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u_1+V_1(x)u_1-\lambda _1 u_1=\mu _1 u_1^3+\beta u_1u_2^{2}+\kappa u_2~\text {in}~ {\mathbb {R}}^3,\\ -\Delta u_2+V_2(x)u_2-\lambda _2 u_2=\mu _2 u_2^3+\beta u_1^2u_2+\kappa u_1~\text {in}~ {\mathbb {R}}^3,\\ u_1\in H^1({\mathbb {R}}^3), u_2\in H^1({\mathbb {R}}^3), \end{array}\right. } \end{aligned}\) under the constraint \(\begin{aligned} \int _{{\mathbb {R}}^3} u_1^2=a_1^2,~\int _{{\mathbb {R}}^3} u_2^2=a_2^2, \end{aligned}\) where \(\mu _1,\mu _2,a_1,a_2,\beta >0,\) \(\kappa \in {\mathbb {R}},\) \(V_1(x)\) and \(V_2(x)\) are trapping potentials, and \(\lambda _1,\lambda _2\) are Lagrangian multipliers, this is a typical \(L^2\) -supercritical case in \({\mathbb {R}}^3.\) We obtain the existence of solutions to this system by minimax theory on the manifold for \(\kappa =0\) and \(\kappa \ne 0\) respectively.
