UUUUUGH! I KNOW WHAT YOU'RE DEALING WITH!! My 5 year old has ALWAYS had problems and now, when he goes, he almost ALWAYS clogs the toilet. We went to visit my mom not long ago and after a week of not pooping, I gave him a children's enema. He went that night and my mom couldn't use her toilet for 3 days. Blocked SOLID. Because not only are their BM'S LARGE, but they are hard. The glycerine suppositories even quit working for him and I have to occasionally give him a children's enema. Usually when we have to force a BM, there's blood involved.
My pediatrician gave me a "clean-out" guide to fully clean him out. One of the main directions was to provide 15 g. of fiber a day. It's hard! Fiber One bars have 9 g. each, but they're expensive. I even tried dried apricots, chewable fiber and Miralax.
The clean-out consisted of daily doses of Ex-lax, which I didn't like at all for a small child. I did do it for the first two days, which resulted in a major blow-out.
The clean-out also recommended Miralax daily in ADDITION to the Ex-lax. Drinks in-between meals, and try to use the potty right after every meal. I would make my son sit on the potty after every meal, even though he said he didn't have to go and more often than not, he would go.
He still has problems but I've found that increasing water intake and physical activity is great. It's hard in the winter time, but we just got a Wii and I have my son do the foot races on the Wii fit., and some tennis and baseball on Wii Sports. About 10-15 minutes and usually that gets things moving and he's actually been regular for the last two weeks. HALLELUJAH! One time, he was so backed up, I could feel a hard lump in his SIDE. Oh, it was awful.
Water, water, water! They even make flavored fiber packets you just add to water- uh..... Crytal Light or something? I can't remember. But I used those also, and they're not too costly. But remember to increase fiber intake gradually as to not cause cramping and gas, and with more fiber you must drink more water.
Good luck! I know how hard it is to make a 3 year old drink water when they don't think they are thirsty! And also remember that for some people, going 3 times a day is normal, and for others, 3 times a week is also OK. My son has been going every other day for the last couple weeks and I am in HEAVEN! They are still large, like, I don't know HOW that came out of my son, but he's not complaining of pain or anything, and he's very proud of himself.