There are thousands of copies of the New and Old Testament, and yet we see that besides grammatical and punctuation errors, there is nothing changed in the New OR Old Testament. Then why do Muslims insist it's been changed? Doesn't it automatically debunk the claim that Muhammad is the messenger of God?
The Qur'an affirms that the Gospel, sent to the Prophet Isa (Jesus), is an Islamic holy book.
Then We sent following their footsteps Our messengers and followed [them] with Jesus, the son of Mary, and gave him the Gospel. And We placed in the hearts of those who followed him compassion and mercy and monasticism, which they innovated; We did not prescribe it for them except [that they did so] seeking the approval of Allah . But they did not observe it with due observance. So We gave the ones who believed among them their reward, but many of them are defiantly disobedient.
It also affirms that Christians and Muslims worship the same God:
And do not argue with the People of the Scripture except in a way that is best, except for those who commit injustice among them, and say, "We believe in that which has been revealed to us and revealed to you. And our God and your God is one; and we are Muslims [in submission] to Him."
However, the modern-day Bible ascribes Allah a son (Jesus Christ), e.g.:
29My Father, who has given them to me, is greater than all[c]; no one can snatch them out of my Father’s hand. 30I and the Father are one.”
The Qur'an flatly contradicts this, describing it as an invented falsehood, e.g.:
They have said, " Allah has taken a son." Exalted is He; He is the [one] Free of need. To Him belongs whatever is in the heavens and whatever is in the earth. You have no authority for this [claim]. Do you say about Allah that which you do not know? Say, "Indeed, those who invent falsehood about Allah will not succeed."
Rejection of the trinity and God having a son occurs throughout the Qur'an. (Consequently, Christians do not accept the God of Islam as their God; see: Is the God of the Qur'an the same god as the God of the Bible?.)
Since the Gospel was revealed by Allah, we expect it does not blatantly contradict the Qur'an, so it logically follows that the modern-day Bible differs from the Gospel revealed by Allah.
See also Why are some verses missing in the NIV? and How many 'Books' never made it to the Bible (Old and New Testament)? at Christianity.SE.
In short, the Bible is really a lot like, say, the New York Times best seller list. It is a compilation of what's "best". What's not included isn't trash- its just not in the same league. The same is true of the canon.
(There's also disputed parts of the Bible, such as Mark 16.)
Muslims do not consider the Qur'an as "compiled" the same way the Bible was. We don't talk about things being "left out" or "included" in the Qur'an: We have not neglected in the Register a thing. (Qur'an 6:38). Instead, we think of the Qur'an as being exactly what Allah revealed, and He protects it: Indeed, it is We who sent down the Qur'an and indeed, We will be its guardian. (Qur'an 15:9) While the Qur'an has various readings (What are the readings (qira'at) of Quran?), the differences are minor (Examples of verses of the Qur'an with two different readings?).
Assuming the Gospel was revealed how the Qur'an was revealed, we wouldn't expect this compilation process.