If a Christian accepts the bible is wrong in places, then they're surely admitting the bible is an unreliable source, therefore it must follow that they should then be questioning whether the story of Jesus is just a fabrication. I find it hard to understand how this part isn't questioned by Christians, when they know the Bible is an unreliable source.
The Bible contains many different kinds of document. It is not a single book with a single author, but a composite of several centuries-worth of text including genealogies, polemics, prophecy, proverbs, wisdom, erotic poetry, history, mythology and theological exposition (among other things). It is perfectly reasonable to accept that some of these texts are likely to be more accurate than others. Some are not even intended to be 'accurate' - can proverbs be
accurate? Can erotic poetry be
accurate? Is mythology or prophecy intended to be accurate, or to make theological and moral points?
Leviticus, as I'm sure you know, contains many rules that God apparently revealed to Moses to deliver to the Jews. These rules include an injunction not to eat shellfish, and details about what you should do if you discover a mildewed brick in your house. A lot of people believe now that these are a mixture of folk wisdom and practical rules for surviving as a people in exile in the desert. It is not insane or hypocritical of a modern-day Christian to reject at least some of these rules on the grounds that they were not intended to be used forever and many are no longer applicable.
Romans, which was written by St Paul, contains one of the most popular passages used to condemn homosexuality. Paul was not Jesus, and nor was he writing a history of Jesus' life. His letters are a theological exposition on the life of Christ. They are, arguably, one person's opinion, and likely to be flawed.
Usually, when history is found in the Bible, it is history seen through the eyes of Christian faith. The four canonical Gospels each contain at least a measure of historical truth, in my considered opinion as an atheist student of theology. But they are not pure histories; they were not written by eyewitnesses, and they also clearly contradict one another at certain points. However, for many Christians this is not the point, because they are also theological documents. For the Christian, there is
faith that the theology of these books is good and right. That means that where the Gospels might compromise on historicity as a means to make a theological point, the wise Christian can evaluate them as they were intended to be understood: as stories written to preach the Good News, and not as blow-by-blow 100% accurate accounts of every detail of Jesus' life.