How about, "I can't understand the kind of English that native speakers use very well."
Though I agree the 'very well' is at the end of the sentence, a long way from "understand.' It almost sounds like the 'very well' means "I can't understand when they speak very well!" You could put the 'very well' closer. "I can't understand very well the English of native speakers."
But if you think that the 'very well' sounds better at the end of the sentence (and I think it does) then you'll have to find a way to say it all. "I can't understand native-speaker English very well." That's shorter, isn't it!
Or do you want to point out which particular bits you don't understand? Idiomatic English? "I don't understand IDIOMATIC English very well." Because it's probably the more informal idiomatic parts you don't understand well.
And OF and THAT both work - but in different structures. ... OF English speakers .... THAT they use.
Example: I don't understand the French of French Canadians/ I don't understand the French THAT French Canadians use.