I wouldn't say given more weight. I'd put it more like anti-abortion laws are simply saying that the fetus has rights, period. If you think that the fetus is a human being, or not quite a human being but some entity that is entitled to rights, then you can't just eliminate it. There's a law against murder, for example. I wouldn't frame such a law by saying that the rights of the person being murdered are "given more weight" than those of the murderer.
After all, pro-choicers (or at least all I have seen) have no problem a person who kills a pregnant woman, even with the baby before the point of viability, being charged with 2 murders. Does the simple fact that the unborn child is wanted make killing the unborn wrong? If the child isn't wanted, either before or after viability, then eliminating it is OK? I don't really follow the logic. If the woman was murdered on the way to an abortion clinic, should the murderer be charged with just a single murder?
I've just never been able to get on board with the "...but it's the woman's body" argument because it...just isn't entirely true. Abortion is not a procedure akin to an appendix removal, a knee replacement, or a nose job, but pro-choicers try to frame it in that language. There is another life/potential life in question here.