There are roughly 30 references to polygamy in the OT. Lamech (Gen. 4:19ff), Abraham (Gen. 16:3), Jacob (Gen. 35:22; 37:2), and Solomon (1 Kings 11:3-6) are just a few examples. The NT authors speak against polygamy (1 Cor. 7:2; 1 Tim. 3:2, 12), but what about the OT? Did the OT writers promote polygamy or prohibit it?
First, description is not the same as prescription. The Bible is a realistic book, and it doesn’t spare any of the gory details. In fact, it describes horrible events such as male gang rape (Gen. 19) and female gang rape and murder (Judg. 19). While the Bible describes these events, it doesn’t prescribe them. That is, while it records them, it doesn’t condone them. This is also true for polygamy.
Second, the original marriage discourages polygamy. In the original creation (Gen. 2:24), God prescribed that a man should have only one wife, one father, and one mother (singular in each case). Jesus affirmed God’s original design for sex by quoting these two passages (Mt. 19:4-5), as did Paul (1 Cor. 6:16).
Third, stories of polygamy always end badly. Whenever we see polygamy in OT narratives, it always brings about massive problems for those who engage in it (e.g. Abraham, Esau, Jacob, David, and Solomon). What do you think God was trying to communicate with these stories? (For instance, compare Deuteronomy 17:17 with 1 Kings 11:2.)
Fourth, polygamy is arguably banned in the OT law. Leviticus 18:18 says, “You shall not marry a woman in addition to her sister as a rival while she is alive, to uncover her nakedness.” This verse is often interpreted as referring to incest, because of the context. However, it might actually be a turning point in the chapter, which refers to non-familial sexual immorality. That is, before verse 18, the context refers to incest. However, after verse 18, it refers to general sexual sin. Copan writes,
This phrase ‘a woman to her sister’ and its counterpart, ‘a man to his brother,’ are used twenty times in the Hebrew Scriptures, and never do they refer to a literal sister or brother. Rather, they are idioms for ‘one in addition to another.’ So this verse doesn’t refer to incest; rather, it refers to the addition of another wife to the first (i.e. polygamy). [emphasis mine]
While this passage is debated, there are other clear passages that discourage polygamy. For instance, Deuteronomy 17:17 bans polygamy for the king of Israel, and Proverbs 5:15-20 gives the ideal structure of a marital sex life (“drink water from your own cistern”). The laws of Israel are not ideal (see “Tips for Interpreting OT Law”), and this could be an example of this.
Fifth, polygamy may have been a ‘lesser of two evils.’ Because men were more rare due to constant warfare, polygamy may have been a lesser of two evils. Without a husband, women were often destitute (e.g. the book of Ruth). God may have allowed polygamy so that these women would be taken care of in some form—even if it wasn’t ideal.
 Copan, Paul. Is God a Moral Monster?: Making Sense of the Old Testament God. Grand Rapids, MI: Baker, 2011. 113. Gleason Archer also holds to this view. He writes, “The basic principle underlying monogamy is contained in Lev. 18:18 for the term sister in that verse may also imply ‘another woman.’” Archer, Gleason L. A Survey of Old Testament Introduction. Third Edition. Chicago, IL: Moody, 1998. 259.