Yup. Think of these possessives as, in a way, adjectival. So:
A red car - Car coch
Dai’s car - Car Dai
It’s just another way of describing a car, and it’s all done through word order. The complication you’re facing is that when using the personal pronouns, a slightly more complex (and variable according to personal preference) pattern is apparent, but you don’t have to worry about that with other possessives.
another tip - the word order may be easier to ‘get’ if you imagine a silent “of” after the thing ‘being possessed’ - birthday (of) my wife, birthday (of) your wife, etc.
The ‘of’ isn’t required in the welsh but it might help to make sense of the word order in translation. The best thing to do though is not worry about the ‘why’ too much at first - it’s one of those patterns that you’ll soon pick up without having to think about it!