There is a lot to get into here, but I’ll give it a shot.

First of all though, I am not sure if the balance ever tipped in favour of violent conversion, at least not by forcing people to convert at sword-point. That was not something that church authorities generally approved of. For example, the Mainz Anonymous, a Hebrew account of the First Crusade, describes how when the crusaders began killing Jews in Speyer in 1096, the bishop arrived in force to protect the Jews. The chronicler notes that because of the bishop's actions, those who were forced to convert were able to return to Judaism. It’s not that the Church had any particular love of Jews, but rather as political as well as religious authorities, the Church did not tolerate the indiscriminate killing of people, Christian or not.

That being said, those crusaders were travelling through Speyer on a Crusade called by the Pope, so there was obviously also a form of church-sanctioned religious violence. The violence at Speyer and really the whole First Crusade is representative of a much wider debate within Christianity about the legitimate use of violence. So, if you would allow me to rephrase your question somewhat, we might ask instead, when did Christians start justifying violence, particularly on the part of kings and states, to spread the faith? How was that violence justified and what were its limits?

While the famous line "all they that take the sword shall perish with the sword." (Matthew 26:52) speaks to Christianity's pacifist origins, the New Testament also sanctions the state's legitimate use of coercion. In the epistle to the Romans (ca. 55-58 AD), Paul writes "For the one in authority is God’s servant for your good. But if you do wrong, be afraid, for rulers do not bear the sword for no reason. They are God’s servants, agents of wrath to bring punishment on the wrongdoer." (Romans 13:4). Readings of this verse would ultimately lead to the development of the notion of the "two swords," the idea that God has ordained two forms of government, the secular and the ecclesiastical. This would lead to a number of conflicts between secular and ecclesiastical authorities in the medieval period, but that's a whole other can of worms.

By late antiquity we also see Christian writers associating the expansion of Christianity with military victories. Eusebius’ (ca. 260-265 - 339-340) famous account of the conversion of the Roman Emperor Constantine in which before battle against his rival Maxentius he and his soldiers saw in the sky, “a cross-shaped trophy formed from light" bearing the inscription “By this conquer.” According to Eusebius, that night Jesus came to Constantine in a dream to clarify that he should display the sign of a cross to protect his army. Similarly, writing about two hundred years later, Gregory of Tours (ca. 538-594) describes the Frankish King Clovis converting to Christianity after asking the God of his Christian wife Clothilda to save him from defeat against the Alamani. These were not wars fought with the aim of expanding Christianity, but we do see chroniclers assigning a divine purpose to warfare. However, while Christian rulers might gain new territory this way, it was not a simple matter to make their new subjects Christian. This was a long-term project that could take decades or even centuries. This Reddit postThis Reddit post gives a little indication of the uneven practice of Christianity even after it was adopted as the creed of rulers.

As Christianity became entrenched in Europe, it also adapted to its martial culture. This was expressed, for example, in the so-called Peace of God, an eleventh-century movement led by laypeople and bishops to enforce an oath on nobles to end their feuding and direct their violence towards protecting the innocent against bandits and the like.

The eleventh century was a definitive moment for the development of a Christian idea of holy war. As J. Riley-Smith has argued in The First Crusade and the Idea of Crusading, while there existed religious justifications for violence and warfare before, Pope Urban II's proclamation of a Crusade to retake Jerusalem in 1095 was a watershed moment. As noted above, there had long been a Christian military tradition. Moreover, popes had called for Christians to take up arms against Muslims before. What made this call unique was that it was a declared holy war on Christ’s behalf that offered remission of sins to all who took up the call. All the accounts of Urban's speech at Clermont calling for a crusade come from after the capture of Jerusalem in 1099, but we can get some sense of the rhetoric used to justify the crusade. Here’s an excerpt of his speech as related by Fulcher of Chartres (who was present at Clermont):

"All who die by the way, whether by land or by sea, or in battle against the pagans, shall have immediate remission of sins. This I grant them through the power of God with which I am invested. O what a disgrace if such a despised and base race, which worships demons, should conquer a people which has the faith of omnipotent God and is made glorious with the name of Christ! With what reproaches will the Lord overwhelm us if you do not aid those who, with us, profess the Christian religion! Let those who have been accustomed unjustly to wage private warfare against the faithful now go against the infidels and end with victory this war which should have been begun long ago. Let those who for a long time, have been robbers, now become knights. Let those who have been fighting against their brothers and relatives now fight in a proper way against the barbarians. Let those who have been serving as mercenaries for small pay now obtain the eternal reward.”

A few points are notable here. First, warfare becomes linked to pilgrimage and penance. Second, notice the goal is not to convert, but rather to simply recover territory lost and support the Eastern Christians against their Muslim enemies. While zealous crusaders would kill and forcibly convert people along the way (in some cases even the Eastern Christians they were supposedly helping), this was not their sanctioned mission or their primary purpose. Third, we can also see as an outgrowth of the sentiments that motivated the peace of God: the vendettas and violence of the nobility can be redirected against non-Christians.

There were a number of factors that led to the success of this particular crusade: the context of the Peace of God, the enthusiastic preaching of preachers like Peter the Hermit (indeed, Pope Urban’s initial call did not gain much traction), millenarian expectations that Christ’s return was soon at hand, and, the division and weakness of the Muslim states at the time. Crucially, it was a success, legitimizing the idea of Crusade, leading to the creation of several crusader states in the Holy Land along with the creation of new military orders which blended religious devotion and martial skill, such as the Teutonic Knights, whose original purpose was to establish hospitals for pilgrims to the Holy Land. The Holy Land crusades led to what I’m going to call “spin-off crusades” that harnessed the same religious fervour to other goals. The Teutonic Knights are therefore much more famous for their role in the Baltic crusades. In the early thirteenth-century the Albigensian Crusade in Southern France directed holy war against heretical Cathars rather than non-Christians.

There is much more to be said - I haven’t even touched on the “Reconquest” of Spain, which ran roughly contemporaneously with the Crusades - but I’ve gone on for quite a bit now and I hope I’ve provided you with the information you were looking for. To sum up, Christianity was quick to adopt religious justifications for war and violence, but it was during the eleventh and twelfth centuries that those ideas really took shape in a lasting way. While full-blown crusades would eventually fade away by the early modern period, the idea that Christianity could be advanced through conquest and empire building endured.