Polygamy was obviously not part of God’s original design for marriage (Gen 2:24; also see 1 Tim 3:2, Tit 1:6), yet it does seem that polygamy was permitted under the Law of Moses and practiced by several famous OT character (e.g. Jacob, David, etc.).
To help us understand why, it is useful to look at the following passage, where the Pharisees are questioning Jesus about divorce:
They said to him, “Why then did Moses command one to give a certificate of divorce and to send her away?” He said to them, “Because of your hardness of heart Moses allowed you to divorce your wives, but from the beginning it was not so.
Jesus shows that whilst divorce was allowed under the Law of Moses, it was not God’s original design for marriage. The key phrase here “because of your hardness of heart”, i.e. divorce was permitted because the Israelites were so stubborn. The same is probably true of polygamy.
One of the important things to bear in mind is that the Law of Moses was not only intended to set the moral standard but also to govern an imperfect nation. Divorce was not approved of, but it happened — there needed to be some laws to govern what to do when it did. If you look at the laws regarding divorce in the Law of Moses, none of them actually establish a law of divorce, they all concern what to do if divorce takes place. In the same way, the laws about polygamy under the law are generally about what to do if polygamy takes place (e.g. Deut 21:15f).