I've heard a lot of people on twitter, reddit, and YouTube arguing for baptismal regeneration. What's shocking is that many of these people call themselves Protestant, even Reformed. I've even heard claims that baptismal regeneration is part of Reformed Theology.
This doesn't make any sense, however. Reformed theology sees baptism as the new circumcision. But circumcision wasn't how people were saved, because if that were the case, then there would be no women who died under the old covenant in heaven. But the Bible distinguishes between circumcision of the flesh and circumcision of the heart. Remember that Abraham was justified by faith before he was circumcised, just like someone like me who wasn't baptized as a child is saved before their baptism. Water doesn't save you, it points to what saves you, which is God's grace given to us through faith in Jesus Christ.
If baptismal regeneration is true, then covenant theology makes no sense at all, not to mention that it contradicts Sola Fide.
I'm Reformed theology, the physical act of baptism isn't what saves you. But the inward reality of baptism - that is, the baptism of the Spirit does save you.
Lutheran here. I'll say something which will probably confuse and enlighten things: I believe that baptism saves... by faith alone. Calvinists tend to see baptism as a symbolic covenant seal, while Lutherans see it as an actual, supernatural working of God (aka a "miracle") to deliver the forgiveness of sins to those who believe (or "the elect" if you prefer). If you believe in monergistic salvation, it makes sense that there should be a specific point in time where God supernaturally creates saving faith in you. Luther argued that this happens in the waters of baptism, since the Word of God is present there with the water
Here's the big kicker: pretty much all of this depends on what you believe concerning irresistible grace. Luther wanted to make it clear that although the action of baptism really does offer miraculous power, that power is only received by faith. In other words, Luther saw no contradiction between baptismal regeneration and Sola Fide. Baptism regenerates, but only for those who believe
Side note: I'm enrolled at a Lutheran pietist seminary and am therefore knee-deep in the tension between the objective reality of God's saving work and the subjective experience of us converting. I'm starting my 3rd Systematic theology course in a few weeks, so I should be able to weigh in on this more after I finish that up