You likely have air trapped somewhere it shouldn't.
Water doesn't compress so when you turn on the tap, the flow is the pressure from the city divided by the resistance of all the pipe between it and the tap.
If you have an air bubble, then when the tap is shut off the air will compress until the pressure matches that from the city. When you turn the tap on, the flow will be the pressure from the air bubble divided by all the pipe between it and the tap. Since there is much less pipe in this case, the flow is much higher.
The pressure in the air bubble quickly drops as the water is pushed out until you again match the original case.
This is especially noticeable if you have have a smaller pipe somewhere between the city and the air bubble than between the air bubble and the tap since obviously the flow is limited by the smallest (cross-sectional-area) pipe.
Depending on the size/shape of the filter, it could be air in the filter itself. See if you can fully fill it with water before connecting it.
If you shut off the house water while making the change, then air could have gotten into any branch going to a higher location than your sink. If you haven't tried running fresh water in every single location in your home, that may be all it takes to clear out the air.
First, the square footage of the house doesn't mean a thing in this case. The important factors are water pressure and supply line size and the height of the fixtures above the pressure tank and the pressure tank size. The well pump fills the tank, then the air bladder applies pressure to the water in the tank. a check valve stops water from going back towards the well. When you call for water, the internal tank pressure pushes it through the pipes.
A few different problems may exist here that would cause your problem.
The tank pressure membrane may be waterlogged. It will still show static pressure, but drops to a very low PSI when a faucet is turned on. Often, you will hear the pump cycle more that usual in this situation or you may see the water pulsate instead of a steady flow. What is your static vs open line pressure?
Another common problem is the backflow check valve is leaking. This causes water to be pushed back towards the well. the tank pressure will slowly decrease between pump runs.
If this problem has been long standing, then the pipe size going to the second level may be too small. Second floor fixtures or ones a longer distance away should be fed with at least 3/4 inch line, not 1/2 inch line.
The last factor may be that the well is not delivering enough gallons per minute. If the pump has to run for excessive time to fill the pressure tank, this could be the problem.
I'm sure there may be a few other items to check, but these are the most common problems.
Best Answer
My guess is there is a restriction in the line going to the house, with only enough getting through to feed one thing, either the toilet or the faucet. And since the toilet is probably drawing the water first before the faucet, it goes there.
I agree that it is more likely just heavy rain that is causing mud in your water. However, surface water getting into your water supply is bad news. You should consider adding filters to clean up the water supplied to the house. I'd guess a chlorine injector to kill any bacteria in the water, a tank to give the chlorine some time to do its job and let mud settle out, coupled with a filter that can take the remaining mud and chlorine out.