You will always hear never leave your child unattended with a dog, any dog. Not just bulldogs. I agree with this scenario for the simple fact that the majority of people have NO idea how to properly socialize, train, or recognize behaviors in a dog. They don't realize when their child isn't recognized as the "alpha" by the dog. They don't realize that an animal is still an animal and should be treated as such. They choose to, and remain clueless forever for the most part.
Now with that said, I will also say that
personally I WILL NOT OWN A DOG THAT I CANNOT TRUST TO BE IN THE SAME ROOM WITH MY CHILD UNSUPERVISED!!! It's a parents responsibility to protect their child, and I am extremely overprotective of mine. Yet, her and the dogs romp the yard together, they play with toys, the used to dig together. I completely trust my daughter and my dogs to be laying on the floor alone watching tv while I'm cooking, or on the computer. I completely trust my dogs to be in my daughters bedroom while she's playing. I'm completely fine with her swimming in her kiddie pool with them laying beside her or her chasing them with the water hose. She respects them, and they respect her. That being said I can trust them to never hurt her, they wouldn't be allowed in my home if I ever suspected otherwise. She is also almost 6, and a very mature little girl. I have not always left them unsupervised either, she had to learn and they did as well.
You have to make your own decisions, if you don't trust the dog or yourself completely.. Then my opinion is NO, supervise them always. It's always better to be safe than sorry though, accidents can happen as they say "in the blink of an eye"...
As a child I ran the neighborhood with my dogs as well, sometimes a whole clan of dogs~ lol depending on the amount of children and who all joined in that day. We ran day and night on bikes, played in woods, you name it with them always close behind. I was never mauled nor did any dog I ever had as a child snap on me either...
By the way Welcome to the site, and Congratulations on your girl, not to mention thanks for adopting!!
