Messed up or otherwise unconventional drama families:
- Baker King - Lots of cheating and lying and abuse, really. Although countered with the Pal Bong Bakery family, which is happy and saccharine sweet, so maybe that's not what you want to see. Maybe just fast-forward through them, lol
- Babysitter - This one's pretty twisted and fun. And short. And on your PTW :)
Dramas with little or no focus on families:
- Awl - There are some smaller arcs involving a mother and son, and sometimes family comes up as motivation, but it's not a major part of the story at all...but you've seen Cart, so you would know what you're getting into, haha
- Once Upon a Time in Saengchori - There is a father and daughter, but they're both fairly nuts; there's also a girl and her grandpa (I think? It's been a while since I watched it...). I guess there's another bit with a dad and his kids that becomes important towards the end, but the vast majority of the drama is pretty family-free
- Taereung National Village - A very rare kdrama that's pretty romance-focused, but has absolutely no meddling parents or grandparents! Just young adults figuring out life and love, although it's less cheesy than I made it sound. Also nice and short.
There are tons of movies that either don't have much focus on family or have more twisted families, so I'm sure someone will put together a better list than I can, but I would include
Set Me Free in the twisted category. It's kind of more or less about a kid trying to escape his family.