Does Torah allow polygamy? The answer is yes, it does!
Before I begin, I need to stress that Torah does not recommend polygamy. Torah states very clearly, that a normal relationship is monogamous, not polygamous. Polygamy is not for everyone.
Since the Hebrew Patriarchs, who had multiple wives, polygamy was an accepted practice. Polyamory, on the other hand, is prohibited.
Why polygamy was allowed? Let's take a look at the relevant verses:
YLT Deut 17:17 And he doth not multiply to himself wives, and his heart doth not turn aside, and silver and gold he doth not multiply to himself--exceedingly.
As you can see, Torah prohibits only excessive polygamy. If you have a few wives it is totally ok.
YLT Dt 21:15 'When a man hath two wives, the one loved and the other hated, and they have borne to him sons (the loved one and the hated one), and the first-born son hath been to the hated one;
YLT Dt 21:16 then it hath been, in the day of his causing his sons to inherit that which he hath, he is not able to declare first-born the son of the loved one, in the face of the son of the hated one--the first-born.
YLT Dt 21:17 But the first-born, son of the hated one, he doth acknowledge, to give to him a double portion of all that is found with him, for he is the beginning of his strength; to him is the right of the first-born.
As you can see, these verses explicitly acknowledge and explain why one may need two wives, namely due to love in this case. In this case, a man marries a woman which he thinks he loves but at the end he does not, so to stay happy, yet not to betray his partner, a man may choose to marry another woman who he loves. And in this case, we see that Torah is on the side of the women and does not imply anything bad at all.
YLT Ex 21:10 'If another woman he take for him, her food, her covering, and her habitation, he doth not withdraw;
YLT Ex 21:11 and if these three he do not to her, then she hath gone out for nought, without money.
As you can see from these verses, even a slave-woman was entitled to food, expenses and housing, otherwise she could become free without paying for herself. Torah shows great humility even to the slaves!
This verse also means that even though it is ok to take multiple wives, you are responsible for supporting these women. So even though there is going to much more sex, there will be much more expenses AND problems as well!
Another point I wanted to make that you can have a threesome with your wives. It is permitted as well. Leviticus 18:22 and Lev 20:13 is only applicable to men because women do not have necessary sexual organs to have sex with each other.
So, why would not be polygamy allowed? If anything, it is natural and normal for human beings. Here in US we allow homosexuals, but not polygamy? Its unbelievable discrimination and misjustice!