  In this paper, we look for solutions to the following coupled Schr\"{o}dinger
system \begin{equation*} \begin{cases} -\Delta
u+\lambda_{1}u=\alpha_{1}|u|^{p-2}u+\mu_{1}u^{3}+\rho v^{2}u & \text{in} \ \
\mathbb{R}^{N}, -\Delta v+\lambda_{2}v=\alpha_{2}|v|^{p-2}v+\mu_{2}v^{3}+\rho
u^{2}v& \text{in} \ \ \mathbb{R}^{N}, \end{cases} \end{equation*} with the
additional conditions $\int_{\mathbb{R}^{N}}u^{2}dx=b^{2}_{1}$ and
$\int_{\mathbb{R}^{N}}v^{2}dx=b^{2}_{2}.$ Here $b_1, b_2>0$ are prescribed,
$N\leq3$, $\mu_{1}, \mu_{2}, \alpha_{1},\alpha_{2},\rho>0$, $p\in (2,4)$ and
the frequencies $\lambda_{1},\lambda_{2}$ are unknown and will appear as
Lagrange multipliers. In the one dimension case, the energy functional is
bounded from below on the product of $L^2$-spheres, normalized ground states
exist and are obtained as global minimizers. When $N=2$, the energy functional
is not always bounded on the product of $L^2$-spheres, we prove the existence
of normalized ground states under suitable conditions on $b_1$ and $b_2$, which
are obtained as global minimizers. When $N=3$, we show that under suitable
conditions on $b_1$ and $b_2$, at least two normalized solutions exist, one is
a ground state and the other is an excited state. We also shows the limit
behavior of the normalized solutions as $\alpha_{1},\alpha_{2}\rightarrow 0$.
The first solution will disappear and the second solution will converge to the
normalized solution of system (1.1) with $\alpha_{1}=\alpha_{2}=0$, which has
been studied by T. Bartsch, L. Jeanjean and N. Soave (J. Math. Pures Appl.
2016). Furthermore, by refining the upper bound of the ground state energy, we
provide a precise mass collapse behavior of the ground states. The results in
this paper complement the main results established by X. Luo, X. Yang and W.
Zou (arXiv:2107.08708), where the authors considered the case $N=4$.
