This is a new issue. We do have a water filter system before going to the fridge, but it was not an issue for the last 4 years. Now, the water come out normal and then quickly restricts and goes very slow. It's causing the ice maker to leak into the fill bucket as well. I think because the pressure isn't hard enough to get into the trays. I've checked everything coming into the fridge. Would the next solution be the water inlet valve?
Sometimes new water filters will discharge a small amount of charcoal or other filter media. It's possible for it to happen any time, but you normally see it with a new filter. It's possible that some filter media has clogged a screen filter just before the water line enters the fridge or the first valve in the fridge. The screens are normally placed where they can be cleaned, so you shouldn't have to dig for it.
Also, the way you describe the problem of the water coming out normally at first and then slows down points to a flow restriction rather than pressure. When not in use, the tubes in the fridge build up to the right pressure so when you first use the water you get a healthy stream, but the flow restriction causes the output pressure/volume to drop.
If the water valve is mounted in the wall behind the fridge, there shouldn't be any damage to the line (at least very unlikely), but if it runs from the nearby kitchen sink, the water line could have been kinked or otherwise crushed.