There are two problems to solve:
1. Break up the power of the flow of water.
2. Make it go somewhere where it won’t be a problem.
Your situation sounds exactly like where my daughter and her family live: On the down side of a steep hill that is not county maintained. Every rain they get washouts...................
But over time the ditch on the hill side filled in with too much leaf debris and successfully blocked the overwash water causing it to veer right into the center of our road. The rain was torrential and even the dense pack could not withstand it.
I've significantly deepened the ditch and will deepen it further, eliminating the halfway point wash off but also fill it with those rocks that are approximately 7" diameter that everyone uses around here for ditches. That should make the fix permanent with minor necessary maintenance. BTW, Here is the hill (in the background). You only see the bottom but the hill (with the tip of the house showing) gives you an idea of how high it goes. And all of that climb is on the front of the hill.
Before we bought it (note the for sale sigh behind our rental car)
From the top (shortly after we purchased it and different rental car):
Notice that in both of those pictures it is just indigenous "gravel and rather bumpy. After we graded it, laid down the dense pack, and I trimmed those trees, it was actually very nice. And when we get the dense pack in, it will be again.