Some people consider that the words of Jesus in Matthew, Mark, Luke, and John, are not part of the new Testament but belong to the Old. They consider that the New Testament teaching can be found in the epistles of the apostles, but not in the recorded words of Jesus himself. They represent Jesus as the last Old Testament prophet who, during his life taught a different message than the message he gave his apostles to teach after his death and resurrection.
We read that Jesus was "a prophet like Moses"(Acts 3:22-24, cf Deu 18:15-18). Does this mean that Jesus was an Old Testament prophet like Moses was? No —Moses typified Jesus Christ and Jesus was the New Testament antitype of Moses the Old Testament type. As the antitype of Moses, Jesus can be said to be like him. But since antitype is greater than type, Jesus is unlike Moses in certain attributes...
The Law came through Moses but grace and truth came through Christ (Jhn 1:17-18).
Christ was the "spiritual rock" typified by Moses who gave the people physical water from a physical rock (1Co 10:1-4).
Moses put a veil on his face because his was a ministry of condemnation. Christ's was a ministry of unveiled glory and liberty (2Co 3:7-8,17).
Christ was faithful like Moses, but Moses was a servant of God as a testimony of those things that would be spoken later by Christ the Son of God (Heb 3:1-6).
After Moses spoke the commandments of God he cleansed the book and the people with animals' blood. Christ after he had spoken his message offered his own blood (Heb 9:11-12,19-20).
All of the above would lead us to expect Christ's teaching to be superior to Moses's teaching. Christ did, in fact, say on occasions, "You have heard that it was said in olden times... but I say to you..."(eg Mtt 5:21-48). In these verses Jesus is basically saying, “the old law told you to do such and such, but I say don't”. It is inaccurate to say that Jesus was giving a “better interpretation” of the law of Moses. He was “contradicting” it (eg Mtt 5:31-32).
Jesus Establishes a New Time Zone
The death of Jesus was going to end the Old Testament law and bring in a new covenant in his blood (Rom 10:4, Lke 22:20, Heb 8:6-7). Why would Jesus preach the old covenant that he was about to abolish rather than preach the new covenant he was about to introduce? The preaching of Jesus is seen by his apostles as belonging to the new time, not the old...
Moses was one of the prophets who spoke "in times past to the fathers", whereas Jesus "in these last days" was the one by whom the gospel was "first spoken"(Heb 1:1-2, 2:3).
The Old Testament prophets were "until John" and "since that time" the gospel was preached. Jesus did not preach “until” John the Baptist, but “since” that time. So Jesus was preaching his new covenant, the gospel. He was not preaching the Law (Lke 16:6, 20:1).
Jesus Understood the Mystery
Here's a closing thought...
The Old Testament prophets spoke things that they did not fully understand even though they made careful enquiry and sought to know about Christ (1Pe 1:9-12) Would he keep it a secret?
The gospel is "the revelation of the mystery which has been kept secret for long ages past but now is manifested"(Rom 16:25-26). No good reason can be given as to why Jesus, who is the very manifestation of the mystery, would perpetuate the mystery in his own teaching.