Your new nature is fully-completed.
Your heart is not growing in holiness because it doesn't have to; it is already as holy as it ever will be. A bucket overflowing can't get any more full. Even among Christians who believe God has made them new creations, giving them his righteousness, there's a great deal of confusion over this issue.
Most Christians think their heart is somewhat holier now with a moderate level of improved goodness, but probably isn't thoroughly, 100%, good. So they mistakenly believe their nature will continue to improve with time.
This view short-changes the biblical view of our new nature. As a Christian, your nature is no longer fallen or in need of improvement: It is as steadfast towards God as Jesus' heart is. Your heart no longer possesses false convictions about God, nor harbors any deceit. Nor is your heart easily mislead or self-centered.
The old heart that did possess false convictions, deceit and mistaken conclusions about life was replaced. Not tweaked, not altered or improved. REPLACED. It's gone.
After surgery, Jesus didn't leave the removed heart just lying around your interior world like a rotting organ left in a trash bin after surgery: That old nature is gone. Flash-obliterated: Burned up by his righteousness.
Your completely-new heart only possesses the noble DNA of Jesus and his convictions.
The real question is,
"Then why do Christians still sin?"
The answer is because the life in our new heart has yet to reach the creases of our mind, our choices and convictions. The process of sanctification has nothing to do with our hearts growing in goodness: Our hearts couldn't be more true and noble than they are now. Rather, sanctification has everything to do with our actions, convictions and relational patterns coming into alignment with that new heart and its goodness.
[There is also our flesh, but that too is no longer your identity. And the Spirit wages war against the flesh, not you.]
Jesus does not give approximations or half-solutions:
Our new hearts don’t simply possess a purity like Jesus had: They possess the actual purity Jesus had, his DNA. Our purity is not an approximation of what Jesus possessed: It IS the purity Jesus possessed.