I represented a company that installed roofs and gutters in NC for 4 years. Your gutters are way too far below the shingles. Judging from the pictures this looks like an older house and you may have some issues with how level the roof line is and perhaps that is why. Ideally, you want the gutters touching the shingles on the high end and sloping down gradually to the downspout but not more than an inch below the shingles. From the pictures, it looks like the water is missing the gutters entirely when it rains hard.
You have half round gutters, which tend to act as a "ski jump" if there is any velocity behind the water flow. The way to reduce this is to use a diverter which is vertical piece of metal bent into a "L" shape with the bottom section about 3/4" wide and the vertical part about 4" wide. It is screwed to the outer lip of the gutter. I don't know if you can buy them, our guys just made them from scrap pieces of gutter. They will trap snow and debris and you will want to clean behind them from time to time. The easiest way is with an old paint brush attached to one of those painters extension poles.
I suggest getting a well qualified gutter company involved. You have half round gutters on open rafter tails which may require changing the brackets and there are a lot of different brackets to choose from. You will want someone who knows what they are doing. I doubt you will be able to eliminate the overshooting entirely, but you should be able to greatly reduce it so that in most storms it's fine.