What you describe is common. If we have a certain "mental bandwidth", in our native language more bandwidth is freed up for creating fluent speech. We don't have to concentrate as hard so we can allocate more energy to speaking fluently. In a non-native language, obviously, we have to devote more bandwidth to thinking of vocabulary, grammar, etc., so our allocation to fluency must go down, and stuttering increases. You may notice this, subtly, in your native language too: the longer/more complex our sentences are, the more chance of a stutter. Contrast saying, "The cat got fed. She's been weird though." with "The cat got fed before you came home, but she's been acting weird ever since." The second one takes more bandwidth, and you're more likely to stutter. Unless, of course, you're one of the people who stutters more in their native language! That does happen. I don't know why.