The short answer is no, the story of Jesus was not written to fulfill OT prophecies. Simply put, it is reasonable to assume that the New Testament ("NT") is a reliable account of the lives and events of Jesus and those who followed him in the first century.
The number and type of manuscripts of NT is overwhelming, and they agree with each other to an exceptional degree. It is reasonable to believe they are an accurate reflection of the original documents. (More details on this later.)
The original NT documents were written by men who spent time with Jesus or were companions of someone who spent time with Jesus. Each letter or book is distinctly different depending on the author and circumstances of the writing. There are variations in writing style and specific details — the same story is told from different perspectives by different people. Details emphasized in one account are completely skipped in a separate account. Some parts are vague and require much study and reflection. Some minor parts are not fully understood even to this day (but nothing of major significance). The letters do not show a completely united, repetitive, consistent presentation of events. If they did, it would be a sign of forgery and/or collusion (the same analysis is used on the stories of multiple suspects facing trial today). The variations actually support the conclusion that they are truthful accounts.
Despite the variations in details, there are consistent underlying messages. If you read all the books in the NT (and even the OT!), you will get a good feel for how all the messages combine to form one lovely whole: that we are sinners saved only by the grace of God, through our faith in Jesus. Yet there is also the message that says we shouldn't use that as an excuse for bad behavior. And the oft-repeated commands to love both God and other people. Taken as a whole, there are no wildly conflicting messages. Considering that the documents were written by several different men, in different places, and at different times, they were amazingly consistent in overall themes. It is reasonable to believe that they were reporting the same events and had adopted the same basic understanding of God's messages to us.
There were many forged letters that surfaced over the centuries which are not included in the present "cannon" of the NT. The forgeries are usually pretty easy to spot — they often contained explicit and detailed supports of a certain Christian sect...and usually were "discovered" after that sect had been fighting others about what to believe. They often include anachronisms — for example, they mention a war that had not happened yet at the time of the supposed writing of the document. In a less obvious way, they usually have a different "vibe" or "feeling" to them that does not seem consistent with the generally accepted books of the NT. When reasonably scrutinized and compared with other documents, the forgeries are exposed as such.
The original documents tell of a man named Jesus who was rejected by Jewish experts in OT prophecy, in part because Jesus didn't seem to fulfill the OT prophecies of becoming a reigning king. Those Jews wanted a king that kicked the Romans out of Israel. But Jesus' reign is in the spiritual, eternal world and is accessible to us now in this life and beyond. But it is not part of the earthly political world and is not an earthly kingdom. If the disciples wanted to falsely tell a story that fulfilled OT prophecy, they would have done so in a more direct and simple way. Instead, in some cases Jesus and disciples quote OT verses that may or may not have been considered messianic prophesies. [A side note here: the people living 2000 years ago were not less intelligent than people living today. It is good to remember that even though we have a lot of knowledge and technology it doesn't mean we are any smarter than any other people in history. To think that the disciples weren't aware of their differences and minor inconsistencies is unfounded. They could have written specific and detailed fulfillments of OT prophecy that would have matched expectations, but they didn't.] In the book of John especially, there are multiple places where the Jewish leaders scoffed at Jesus and his disciples, rejecting their claims of prophetic fulfillment. If the NT authors were lying, why include their own failures at persuading the religious leaders? Why not construct better lies? The simple answer is that they weren't lying.
Some Jews expected two or more Messiahs, in order to fulfill both the "Suffering Servant" and the "Reigning King" aspects of the Messiah. They did not expect one man to fulfill both extremes. Yet Jesus did — he suffered and died, then rose again, to bring to option of peace to all who trust in Him. Then, as now, people often aren't interested in that kind of peace. They want everything now, here, without effort or struggle. They wanted multiple messiahs so that at least one of them would be the conqueror in an immediate political and physical way. Jesus didn't meet those (misguided) expectations. He didn't try to appease those who wanted to put him in one or the other boxes.
For many years before and after Jesus, there were men who claimed to be the Messiah. They rebelled against the Roman soldiers, rallied the Jews, and fought for political freedom. None of them brought lasting freedom from oppression. Jesus was unique in his approach. He never became boastful or prideful, never used his position to abuse women and children, never tried to turn his followers into a militia. In more recent times, men have claimed to be the Messiah or Jesus in his second coming. They break God's commandments for living, take advantage of people, take their money and their wives, abuse children, or take up arms to fight governments for political change. Their behavior always betrays them and shows the falseness of their claims.
When examining the New Testament documents, it is reasonable to believe that they consist of a variety of accurate first-person accounts of the life and teachings of Jesus. Claims to the contrary are possible, but not reasonable. We must always examine claims about the bible, whether they come from atheist websites or pulpits in churches. Check for yourself to see if what I have said is reasonable. And feel free to ask any follow-up questions or requests for clarifications. It is wonderful that you are asking these questions, and I thank you for the privilege of responding. May God bless your search for truth and give you grace and peace as you explore His Word.
MORE ON BIBLE DOCUMENTS
It may help you to think of the Bible as a Library, not a book. It is a collection of letters, histories, poems, songs and essays. It was compiled gradually over about 1500 years, and it has over 40 different authors that we know of. The establishment of one specific group of texts to include in the Bible was a gradual process. Before the official "church" decided on which to include and how to arrange them, Christians had long considered certain letters and texts as true, others as false.
Early in my walk with God, I wondered whether I could trust that the bible was true. I did some research. At first I read books by Bruce M. Metzger (probably the most respected New Testament scholar of the 20th century). Dr. Metzger's books are well worth the effort to read them, but I'll warn you the material can be quite dry. I've also read books and watched videos by Daniel B. Wallace (one of, if not THE, current authority on NT manuscripts). Dr. Wallace's videos are far from dry — he is an enthusiastic speaker, so his videos might be a better choice for you depending on your interest level and available time to study. I will try to summarize both men's messages like this:
There is abundant evidence that the New Testament manuscripts are nearly 100% consistent, accurate, and well-preserved. The number of manuscripts and cross-references in other literature are "an embarrassment of riches" (as Dr. Wallace puts it) — there is way more to support the Bible than any other old document we have. There is plain (physical and textural) evidence that the apostles and others who knew them wrote the New Testament. Statements to the contrary are usually ignorant of the facts. Since we can believe that the texts are legitimate and accurate, we can believe that the authors were people who actually knew Jesus or interviewed people who did. Since we can trust the texts, and therefore trust the authors and the information they portrayed, it is reasonable and rational to believe that the New Testament is a reliable source of information about Jesus and his impact on the world. Yes, faith is required to believe anything. But the Bible has lots of physical evidence to support its authenticity.
After I felt confident that the NT was true, I did some reading on the history of the OT. I decided to accept that the Old Testament was also true. I based this in part on the quotations of the OT by Jesus and others. The Dead Sea Scrolls showed that the OT was practically unchanged for thousands of years. So it seems reasonable to me that the OT Jesus and his disciples quoted is the same OT we have today. In short, I learned to completely agree with 2 Timothy 3:16-17 that "Every scripture is inspired by God and useful for teaching, for reproof, for correction, and for training in righteousness, that the person dedicated to God may be capable and equipped for every good work."
If you do the research and ask God to guide you, I believe you will come to the same conclusion. I encourage you to research this on your own. Many popular ideas (spoken by Christians and non-Christians) you hear about the bible's history are false or distorted truths. It is important to know that you can trust the bible to be true and accurate. Knowing the truth about the history of the bible should help you feel confident and dispel doubts. Eventually, we all need faith to strengthen our knowledge and we need love to help us learn and share with others. (Remember "Knowledge puffs up, but love builds up" from 1 Corinthians 8:1.)
Do not just Google your question and believe anything you watch or read. Find the experts and see what they say.
a. A great overview by Daniel Wallace
b. Search YouTube for Wallace's: "Is What We Have Now What They Wrote Then? Part 1 — Biola Chapel". There is also a part 2, and combined the whole thing is under an hour.
c. If you want to do some reading on this subject, start with Chapters 6 and 7 of Reinventing Jesus
, co-authored by Wallace.
d. If you have a lot of time to study this subject, I recommend The Text of the New Testament: Its Transmission, Corruption, And Restoration
, The Bible in Translation
, and The Canon of the New Testament
all by Metzger.
May God bless you as you continue your search for Truth!