Severe toddler constipation

My 15month old daughter has horrible constipation. She cries uncontrollably, she often shakes due to her degree of pushing, and she rarely ever passes a bowel movement. When she does it usually take about a few hours for her to completely pass it with the first few tries of failure. She gets so worked up: she's fine for half an hour then shes screaming, no success after about 20 minutes of crying, then shes fine, and then back at it again. Her bowel movements seem so painful to watch her strain and sometimes there are drops of blood in her dry and unusually large bowel movements. We had been giving her milk of magnesia for few months.. helped her to large extent.. but can't give her everyday.. is she lactose intolerant? Please help with suitable meducine