I'm the same as you - however i've had the constipation even before going on iron tablets....
My doc suggested to me pear and apple juice so i have started this week on that. Prehaps you could try that.....
Again this is tmi... lol.. gota love these topics haha....but try a different position on the toilet rather than sitting. i.e squat on seat to start with; or even stand up to start with.... weird I know... lol and the first few times its usually just wind... however they seem to be more appropriate positions to get things moving.

haha i'm going to leave now