A diaper on a baby should be tight enough to prevent leaks and sagging too much but not too tight that it leaves marks on your babies waist and legs.
A diaper is fitting good and not too tight when you can get 2 fingers under the diapers waistband easily and if not then the diaper is too tight.
The diaper should be loose enough to allow the baby to move freely and not be restricted but you don't want it too loose or it could leak and fall off the baby.
Tight diapers on your baby or toddler should not cause constipation.
However when the diaper is too tight then it can cause the baby or toddler to become uncomfortable cause rashes and red marks around the child's waist.
Tight diapers can also make it hard for the baby or toddler to move around especially if they are walking it can prevent them from sitting down or bending over easily.
So the diaper should not be too tight anyway and if the diaper fits tight then you may need to change diaper sizes and go up a size larger in diapers.
You should be able to fit 2 fingers easily in the diaper waistband between the diaper and the child's waist.
Your baby or toddler may not be getting enough fluids which can lead to constipation.
Most cases of baby constipation or toddler constipation go away on their own within a week or so but if not you should take the baby or toddler the doctor.