There was definitely that throwaway line in "Firstborn" that implied Kurn had a family. My in-universe guess is that when Kurn lost it all, they probably left him as well, and when he went to Worf he had nothing and no one left. I doubt they were killed/murdered, though. Klingons are violent but they have a code they live by as well.
In reality, I think as a whole, when the writers/producers decided to bring Worf to DS9, they didn't want to deal with his "baggage." It was the early days of the internet so I don't they thought we would piece it together... but I definitely did. They established certain things about his character in TNG that didn't seem to fit what they needed from Worf on DS9 so they dismissed it.
Like with Alexander... by the end of TNG it was established that Worf was working on his relationship with his son and things were much better. He even adamantly refused the idea of Alexander not living with him ("Firstborn"). Yet when DS9 starts, for some mysterious reason they are estranged and he lives with his parents, the same people Worf said were too old to care for him back on TNG. My guess is that because they already had a single dad on the station, they didn't want another one. But then when Worf was about to get married they couldn't keep ignoring his son, so they brought him back in a way that fit the plot more than it fit the character. I personally think Alexander running away to Starfleet would have been more of a fit for his character.