How did Christianity get from being a splinter branch of Judeaism to being the dominant religion on four continents and a sizeable proportion of the other two?
Christians completely wiped out Native Religions with more effectiveness than the Muslims did. Which is why you'll still find Zoroasterians in Iran, but you won't find a worshipper of Odin anywhere.
In answer to the question, good salesmanship combined with an appealing product. Empire of Rome making Christianity the only and official religion of the Empire might have helped as well.
Dominate in Europe and North America until recent decades given the numbers leaving the "church" and claiming to not be Christian. Likely the same for Australia. South America might still find it dominate. Asia and Africa will find it running further back in the pack of many other religions.
With the recent rise in paganism, likely may find worshippers of Odin here and there. Zoroastrians are rather scarce, and a bit oppressed. Few Native Religions have been wiped out, though many may have seen their followers change over.
Comparing effectiveness removing/suppressing other religions by Christian versus Muslim, and first point to recall is that there is no text in the New Testament commanding Christ's followers to use violence to convert or kill non-Christians.
Islamic liturgy is another matter.;
Fight those who do not believe in Allah and the Last Day, nor comply with what Allah and His Messenger have forbidden, nor embrace the religion of truth from among those who were given the Scripture,1 until they pay the tax,2 willingly submitting, fully humbled.
(29) Fight those who do not believe in Allah and the Last Day, nor comply with what Allah and His Messenger have forbidden, nor embrace the religion o...
quran.com
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It was narrated from Jabir that the Messenger of Allah (ﷺ) said:“I have been commanded to fight the people until they say: La ilaha illallah. If they say: La ilaha illallah, then their blood and wealth are protected from me, except for a right that is due from it, and their reckoning will be with Allah.”
Hadith | Sunan Ibn Majah | عن جابر، قال: قال رسول الله صلى الله عليه وسلم: " أمرت أن أقاتل الناس حتى يقولوا: لا إله إلا الله، فإذا قالوا: لا إله إلا الله، عصموا مني دماءهم، وأموالهم، إلا بحقها، وحسابهم على الله "
hadithprophet.com
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It was narrated that Abu Hurairah said:"The Messenger of Allah (ﷺ) said: 'I have been commanded to fight the people until they testify to La ilaha ill-allah (none has the right to be worshipped but Allah) and that I am the Messenger of Allah, and establish regular prayers and pay Zakat."
Hadith | Sunan Ibn Majah | عن أبي هريرة، قال: قال رسول الله صلى الله عليه وسلم: «أمرت أن أقاتل الناس حتى يشهدوا أن لا إله إلا الله، وأني رسول الله، ويقيموا الصلاة، ويؤتوا الزكاة»
hadithprophet.com