I understand the misconception, however many times with physics, what one observes is not actually what happens in nature.

This case is answered by the fact that objects in motion MUST BE SLOWED DOWN TO A STOP. Things have to get slowed down, or accelerated, which brings them to a stop, whether it is a bullet hitting a wall or a car's brakes.

Consider a bullet, as you mentioned, being shot at a thick wall of metal. When we observe this, we conclude the bullet will do one of three things. The bullet will either hit the wall and bounce off, go completely through the wall, or, as you suggested, get stuck in the wall. For our example we will consider the third case.

If a bullet is shot into a wall and stops, we know it cannot stop instantly. Once the bullet enters the wall, a large force pushes against the bullet in the opposite way. This force accelerates the bullet in the OPPOSITE DIRECTION, making the bullet slow down until it stops. This force is the reason the bullet gets stuck in the wall.

What if the bullet went completely through the wall? The bullet experiences a force in the opposite direction in the same way, slowing the bullet down. However, what is different about this case is that the wall is not as strong, and doesn't accelerate the bullet enough to stop its motion. There is nothing that says the bullet HAS TO STOP, as it could just be slowed down a little bit. This is why a bullet would not possibly go from its initial speed to a stop instantly. Since acceleration is applied over time, there is no way to accelerate something in an instant, which is one moment of time. Hope this answers your question!

Physically it doesn't make sense if the speed of the bullet goes from 1000 to 0 directly. When the bullet hits the wall, the velocity will drop instantly. The whole process could take less than half a second, or even shorter. It is the scale that you should keep in mind. Suppose you have a supercamera that can record 10000 frames per second, when you play back the process slowly, then you should be able to resolve the drop of velocity of the bullet, going from 1000 to 999, and then eventually to 0.

I like to think about this problem in two ways. First, when the bullet hits the wall, either the bullet, the wall, or both, deform. For example, if the bullet is harder than the wall, it will penetrate into the wall, and slow to a stop somewhere inside. If the wall is harder than the bullet, the bullet will be squished as it hits the wall, so even after the tip of the bullet hits, the rest of the bullet is still moving toward the wall, but quickly slows and stops. In both of these cases, the bullet does not instantly stop.

However, what if neither the bullet nor the wall can deform? Or, in the simplest case, what happens when two atoms collide? The key idea here is that when all matter (i.e., atoms) is pushed close enough together, it repels. And, the closer together the atoms are, the more strongly they repel. So, when an atom collides with another atom, they don’t immediately “hit” each other. Instead, they begin to softly repel, followed by a stronger repulsion as they get closer, as though there was a spring being compressed between the atoms. So, since any collision can be thought of as a compression of an atomic spring between atoms, no collision occurs instantaneously, but instead over some (very small) time period.

The bullet does not stop the instant it hits the wall. If you look at the wall after the bullet has hit it, you will notice that there is a bullet hole in the wall. This is because the bullet continued some distance into the wall (the depth of the hole) before stopping. While the bullet was forcing its way into the wall, it was being subjected to tremendous force that was slowing the bullet to a stop. In the extreme case, the wall actually stretches slightly as does the bullet, resulting in the bullet ricocheting like a bouncing object (which is exactly what it is).