Not an uncommon problem.
Incorrect drip edge gutter installation.
There are two reasons for water getting behind the gutter.
One of the reasons is that your shingles may be too short.
On the eave side of a house such as this i install a 3 x3 piece of flashing to slide the gutter under.
Rain gutter installation can be a tricky job and a lot of.
For proper water drainage the drip edge on the rakes should always overlap the drip edge on the eaves.
Make sure the drip edge coming from the shingle is bending against the fascia board because when it rains it s going.
With what you have there you d probably be better to install a strip of metal under the drip edge and over the gutter.
Home centers sell rolls of 6 in.
A gutter apron is installed on the corners above roof decking.
Use tin snips to cut the gutter apron flashing roll in two 3 in.
While on the other hand a drip edge is installed to the trim and usually hangs directly behind the back of the gutter.
It should never be installed outside the gutter i e against the fascia board.
Not sure if there s a name for it but i call it drip check.
The guttering should run a few inches lower than your roofline.
If there s a drip edge installed where the fascia meets your shingles and the gutter is hung below the drip edge get some roll flashing and tuck it up under the drip edge and over the top of the gutter.
Most premade drip edge flashin.
The whole purpose is to direct the water into the gutters.
This works out correctly if you do the eaves first.
A gutter apron is used where shingles are not far enough from the edge of the gutter.
It is put under the shingles of an attic to flow water directly into the gutter.