I just wanted to second the Miralax suggestion. We had the same problem with my daughter from age 2.5 to 4. The Miralax worked wonders. I mixed it with her sippy of water in the morning - can't even taste it. One other suggestion (but don't know if your daughter is too little for this) At one point when my little one was about 3 I was so frustrated about the whole thing and wanted her pain to go away. I remember sitting her on the potty, looking into her eyes, and very FIRMLY telling her she had to to let it out. It may sound crazy, but I think it got through to her a little. Not sure if you can reason with a 19 month old though. Also, my brother (a doctor) suggested putting her on the toilet backwards. Another crazy sounding idea, but it forced her to really spread her legs, and made it very difficult for her to withhold. Once again, this suggestion might not work for such a little girl, but if your problem persists it might be worth a try as you begin potty training. I hope she's "cured" before that though!
Oh, and one other thing. As far as the diet goes, when my daughter started going through this I was very insistent about her eating a lot of fruit (she's not crazy about veggies) To this day I'm in the habit of cutting up an apple, putting it in a tupperware and taking it with us wherever we go. Especially if we're going to a restaurant where I know all she'll eat is grilled cheese and fries. She eats her apples while we wait for our order to arrive. That might sound silly too, but hey, makes me feel better that I'm getting as much fiber into her as I can!! She even asks "where are my apples?" now. Good luck, A..