I think it's based on two things: where you are and how the letters start when written.
When writing letters n and U, both are started at their leftmost position (for most people) and their topmost position (for most people).
In the UK, or other places where you drive on the left, a u-turn would make a little n. While you aren't starting the turn from the topmost position of the letter, you are starting on the left of the letter.
In the US and other countries where you drive on the right, you make a U-turn because viewed from above you are making a U that starts from the top and left of the letter. While from your immediate perspective while driving your making a lowercase n but starting from the right, from above it's the classic U movement that occurs.
Why would it be an n-turn? The n shape suggests you back up before turning. The capital U asserts that you simply turn. That's how I've always done it.
because it was originally called a You-Turn, because driving instructors said "Now you turn", and people heard it as "Now you-turn", and then abbreviated it as "u-turn"
As humans we are biased towards action and forward thinking. From the perspective of where you are heading after completing the turn, it was a U shaped turn
To deepen that: does a U-turn become a n turn when you have to reverse briefly because the curve was too small and thus give your path a little uptick? n
In most fonts the uppercase U has no extra line, most don't write one in upper case as well when handwriting.
Must be thought off in the US, as they have space on the road for U turns though. Most other countries are stuck with 3 point turns. (Or in my case, a lot more, my car has a turningcircle of a battleship)
I believe OPs point has to do with the direction. From a top-down perspective driving forwards would look like driving upwards, then turning and driving downwards, like an n. U flipped/rotated 180 degrees.
Vehicles vary greatly in turn radius, and it's not just size. I've had big vehicles that could do a U turn just about anywhere, but that damn Saturn...how could a small sedan be so bad at tight turns?
I honestly think it's just because U turn is easier to say than n turn. Because U is a vowel, it doesn't require the glottal stop that's in n(stop)turn.