The way the text of the sermon on the mount reads means that Jesus was explaining, rather than improving on, the Law - by means of correcting the Jewish leaders' misapplication of the Law.
The Law was love in action. If we want to know what the Law meant, look at the life of Jesus - because He did what it says.
That's why Paul also said that Love is the fulfilling of the Law. Love is not an improvement on the Law - Love is the Law and the Law is love.
Jesus' comments were against the wrong way the rulers were using the Law - not against the Law itself.
I don't think we can find anything Jesus taught which actually broke the Law.
Jesus was able to make a case for everything He did and taught using the Old Testament Scriptures as His moral basis for doing so.
There were two ways in which the Jewish leaders were misapplying the Law. One, they were wrongly using it to justify their own immorality. Two, they were wrongly making it harsh on others.
But even if Jesus was improving on the Law, it certainly doesn't mean Jesus was now saying that adultery and sexual immorality is no longer a sin. If anything, He was upping the ante.
Even when Jesus forgave the woman taken in adultery, He was not breaking the Law. The Law required that a woman could be stoned to death only at the testimony of two or three witnesses and that the witnesses had to be the first to cast a stone at her. But no witnesses remained to testify against her.
Therefore Jesus, being the only remaining man in the room, even if He had been an eye-witness (which He wasn't) would not have been required nor would He have been permitted by Law to condemn her. Jesus showed mercy without breaking or improving upon the Law! He demonstrated perfect compliance with the true spirit of the Law - which was both justice and mercy, not one at the expense of the other.
The leaders' hypocrisy was exposed. Where was the man who was found committing adultery with her?
As for the woman, Jesus said, "Neither do I condemn thee...go and sin no more." He forgave her, but He never changed the fact that adultery is still a sin.
It's one thing to extend God's grace to sinners - it's another thing to say, as many are saying, that God no longer actually thinks there's anything wrong with sexual immorality.
I don't know firsthand how modernday "Pharisees" are treating homosexuals, but it sounds bad. As for me, over the years I have led female and male prostitutes in prayer to accept Jesus and seen them touched by the power of the Holy Spirit.
When sinners drew near to Jesus for to hear Him, I don't imagine He was saying things like, "Well you've heard that adultery and homosexuality are wrong, but I say to you it's okay." No - Jesus said He associated with sinners because they needed repentance. And because they were repenting, He said they would enter the Kingdom of God before the Pharisees!
One woman did not cease to weep and to wipe His feet with her tears. That was repentance. That's the effect that truth, combined with love, has.
But being harsh crushes people's spirit. And allowing immorality makes us equally guilty. But truth and grace is what Jesus was all about.
We should show God's love to all sinners. But many today are actually saying there's nothing wrong with continuing in sin. That only keeps sinners in the terrible bondages which they'd like to be free from.