Japanese dramas are wayyy better!

1. Less episodes = less dragging, fast paced and no filler episodes.
2. Stories are usually better and more realistic and just different.

Etc.

I just don't have the patience to watch Korean dramas. Just tried to watch Legend of the Blue Sea...gave up after 20 minutes. So slow and childish. Age of Youth is another so called gem...eeeh NOOO! Watched the first episode and don't feel like continuing it. Will try Goblin now...doubt that will be a succes. 1,5 hour episodes LOL
^ Forget about Goblin. I refuse to torture myself any longer. Going to watch Alice No Toge and Le Uru Onna. Both 10 episodes, 45 minutes. Feels much more relaxed! 
I find a lot of j dramas to be based in a high school setting (pretty boring, plus I really don't want to be reminded of my hs days....*shivers*)  and unrealistic acting. However, I do enjoy the more 'darker' theme/thriller dramas and movies the japs have to offer.

Although,  I'm a sucker for romantic, comedy and action dramas, so I do prefer k dramas. 
Well I tend to watch jdramas more than k-dramas. But it doesn't mean I hate k-dramas. But just like you, I prefer shorter and more concise episodes/stories. Kdramas have a lot of unnecessary scenes or characters just to make the story longer, I guess they do that so they the drama gets more sponsor, etc. I really don't know. Nowadays, Kdrama are pretty popular, and more advertised than jdrama, and that makes me have a high expectations of a particular kdrama once it gets more hyped, and then when I check it out, I tend to be rather disappointed than satisfied. But there kdramas that also met my expectations about them. 
K-drama is a trend now. That's all.
I prefer jdramas, they're more lighthearted
So firstly, whether you watch a K-drama before you judge or not, whatever.  you are closing yourself off from widening your experiences.

As to why K-drama is more popular, there's actually historical and financial reasons for this.

The first free trade agreement in the world based on culture was signed in the mid nineties between Thailand and Korea.  Basically the Thai govt was sick of this new trend the Thai kids had with watching Korean drama over their own locally produced stuff.  This agreement was worth about 50M dollars.

The second free trade agreement in the world based on culture was signed in the late 90s between California and Korea.  From memory, and I could be wrong, was worth about 500M dollars to Korea.  

The export of the culture world wide is now worth billions of dollars to the Korean government. The Korean government realises that export of culture, that is: pop drama, grows the economy and so it invests in the arts.  

This makes it much easier to get funding for projects.  Having government support for modern arts is rare in the world.  This is why Korean drama is more popular and prolific.
Someone mentioned jdrama are lighthearted, well I, and possibly many others, find jdrama are too lighthearted. Other than the "survival of the fittest" or "terminal illness" types, the rest is not really dramatic enough for me, or maybe I'm just not finding the right ones, anyone care to recommend?

It might be just the case that I don't have an affinity for japanese culture and therefore all the drama feel boring but really want to give it another chance. I also think I don't like japanese acting, it feels too stale, and at worst, unrealistic, even if the stories are very realistic.
They're more difficult to get invested in. Korean dramas will almost always have a romance angle - and while it may be cheap and cliche to some (like me) - it's a good way to develop characters and get viewers invested in the characters (and show). It also provides an extra dimension to the show.

Actors and the characters they play can sometimes help mask a weak storyline but when most Japanese dramas seem focused on the storyline at the cost of character development - there is no hiding the weak storyline. There are a few other things I could list as being "problems" with your average Japanese drama compared to average Korean drama but I feel like my post is becoming too long and it would feel like an angry rant.

I think JayRandom might be on to something with that second paragraph. Maybe it is more cultural. If I had to pick - Korean TV/Movies seem the most "Western".

I do watch and enjoy Japanese dramas but I think it's easier to watch a Korean one and enjoy them more.