As a Korean who spends a lot of time on the internet, I would say there is no such website, unfortunately. At least, young people will not like the jokes in that kind of sites. Jokes were in such form in 15~ 20 years ago.
But ever since pictures, clips, musics, drawings, and cartoons are all over on the internet, one-liner or pure text jokes never came back.
For funny stories, search for 컬투 사연 베스트. "cultwo show" is the most famous radio comedy show in Korea. They introduce listeners stories. And they DO know how to tell funny stories. You may find some of them with subtitles on Youtube.
I recommend searching for 웃대 답베 = 웃긴 대학 답변 베스트 = Humor university(name of a community)'s best replies. Usually captured and spread over the internet. Most people find it funny but small number of people may find it too aggressive or dirty. You will need to read main article and also all the replies to get the best reply's point. It is not to get the idea at first. Actually most of old Koreans will fail to understand the jokes. But, these are best combinations of all the current issues, interests, gossips, seasoned with tons of sarcasm (and dirty jokes some... many.. times)