Speaking of Christianity...we don't know. I just want to get that out there. We don't know for sure, our numbers are speculative (Stark was originally a sociologist of far more transparent modern religions who applied his expertise backwards, which poses new challenges).

We have no sure numbers on how many Christians there were. I've heard high ends for 10% during the time of Constantine ( Stark) and down.

In The Rise of Christianity Rodney Stark suggests a growth rate of 4% a year(40% a decade).

Under this view, Christians would have had to convert people but not as much as you'd think. Especially since families may convert together and there was still natural growth.

Perhaps a significant element is something we don't really consider because it's so natural to us: Christianity was both missionary and exclusive.

Pagans accepted the existence of other gods, they didn't need to convert others away from them to the "right" god. Cults were transplanted and gained new adherents but people were still part of "paganism" (however you put that). So you have three things dovetailing that are bad for paganism:

1. Pagans are open to new gods (though they are often not framed as new- certainly the Christians didn't claim to worship a new god)
2. Christians are more than happy to tell them about it
3. Christians are themselves are inured to worshiping other gods (since this is anathema to monotheism which has had centuries since the rise of Judaism to develop tools to resist this).

As MacMullen puts it in Christianizing the Roman Empire the numbers spell disaster for pagans at Sacrifice:

No, the unique force of Christian wonder-working that does indeed need emphasis lies in the fact that it destroyed belief as well as creating it—that is, if you credited it, you had then to credit the view that went with it, denying the character of god to all other divine powers whatsoever.

Let us suppose (if the illustration does not seem too artificial) a hundred pagans confronted with competing miracles wrought by the holy men or holy means of some pagan deity and of the Christian deity; and let us suppose the miracles to have been wonderful in the same degree, and to have produced the same number of converts—fifty each. Yet in the end only the pagan ranks will have been diminished. Fifty pagans have simply added one more name to the pantheon they venerate; but the remaining fifty now deny that pantheon entire. It was this result, destruction, that non-Christians of the time perceived as uniquely Christian; and it was this result which in turn gave so grave a meaning, from the pagan point of view as well as the Christian, to the successive waves of persecution. They were so many waves of desperation

It seems anticlimactic (to me at least) but it's a pretty elegant answer for the question of how it happened and why the Christians continually had issues with pagans (even when pagans were more numerous).

Their evangelizing and exclusive monotheism may simply have had a natural advantage here.