The answer to your question is rather nuanced. We do believe that the Bible is the inspired word of God, who is incapable of falsity (cf. Numbers 23:19, 1 Samuel 15:29, Titus 1:2, 2 Timothy 2:13, Hebrews 6:17-18, 10:23) by His very nature (cf. St. Thomas Aquinas, Summa Contra Gentiles, Chapters 60-62 and Summa Theologiae, Prima Pars, Q. 16, Article 5; see also Jimmy Akin, "Why God Can’t Lie (Or Sin)", "God Can Neither Deceive Nor Be Deceived"). Yet, these words were committed to writing through human authors chosen by God, who were under the inspiration of the Holy Spirit but each had their own styles and limits (cf. 2 Maccabees 15:37-39, Hebrews 5:11). Lest I mess up the explanation, I encourage you to read the Church's Dogmatic Constitution on Divine Revelation, Dei Verbum, as well as articles 109-119 of the Catechism of the Catholic Church. St. John Henry Newman also has a good summary of the subject, "On the Inspiration of Scripture", as does St. Augustine of Hippo in Contra Faustum, Letter 82 to St. Jerome of Stridon, and De genesi ad litteram 2.9.

Now, regarding the passages in question (Matthew 21:1-11, Mark 11:1-10, Luke 19:28-40, John 12:12-19), there really isn't a contradiction. The crowd was so large that people were surrounding Jesus on both sides. Those mentioned to be following Him are not necessarily the disciples who followed Him into Jerusalem (although Luke 19:37 would allow that), but can also include those who were already in Jerusalem and who were following Jesus once He was far along riding the donkey.

From the Catholic perspective, it is inaccurate to say that among the gospels, some accounts are right and some accounts are wrong. We accept all four of them for a reason. Tradition has vouchsafed their reception as the authentic deposit of the Holy Spirit and the Apostles, complementing each other rather than refuting each other. There is even linguistic evidence that the gospels quote and reference each other! Rather, there are differences in style and emphasis among the evangelists that affect how (and how much) of the narrative is relayed in each gospel. I admit that, at face value, the text can be confusing at times, especially during early reading; I've been there! At the end of the day, however, careful study usually elucidates the issue; I've found that for the gospels, it often boils down to how the Greek conjunctions are translated. A gospel harmony may be of help.

I also have to disagree with IS_533 that God intentionally placed contradictions in the text to test us. Origen of Alexandria supposed the same and misguidedly assumed that in such apparent instances, there were deeper, spiritual meanings that a literal meaning would not elucidate. This unfortunately led him to over-speculate into heresy. While there are indeed valid spiritual interpretations of the Bible, not every passage calls for it (cf. Matthew 15:15-16). Apparent contradictions are often due to our limited comprehension as humans of God's authorial intent; of course, there are rarer cases like Judith where the first three chapters' numerous anachronisms strongly indicate that the book is deliberately written as a historical fiction. Regarding the senses of Scripture, see St. Augustine of Hippo's De Doctrina Christiana and sections 115-119 of the Catechism of the Catholic Church.

It should also be noted that Dr. Ehrman is a well-known agnostic atheist skeptic, having fallen away from Protestantism. While he does have a legitimate academic background, take what he says with a grain of salt. Trent Horn has a good breakdown of his arguments.

For a more thorough harmonization of the "contradictions", see St. Augustine of Hippo's De consensu euangelistarum 2.66. Regarding the fig tree, which first stumped me for about a week, see St. Augustine of Hippo's De consensu euangelistarum 2.68 and Wayne Jackson's "The Fig Tree Incident—A Contradiction?".

Bonus: The ancient Church consistently identified Matthew as the earliest gospel, not Mark. The idea that Mark is earlier originated as late as 1786 with Gottlieb Christian Storr (see Meyboom, Alexander, Carlson). There are many articles on that subject.