Depends on the type of pram I have been driving round in a mk6 fiesta tdci my daughter is six months old now, I had no problems I had a travel system so the car seat clipped into the pram frame, the changing bag I had was a slim bag which I could keep every item plus there was a nappie mat neatly tucked away.
So all depends on your equipment but I didn't have any issues. ......
But then I had a m8 who used to struggle with a bmw 320 estate. .