My first reaction to your post is that 5 is awfully young for formal riding lessons.
MOST children aren't physically, emotionally or developmentally riding for a true formal riding lesson until around 7. The don't understand the link between action and consequence, don't follow directions well or thoroughly enough and don't have sufficient impulse control. There are some exceptions, but not many.
If a child is taking riding lessons before 7; the key is to have the lessons be short and frequent. 20 - 30 minutes 3 - 5X a week.
Even if your child is capable, and the instruction is good and safety minded, I would still queston whether this is a good investment. Your child is going to be marking time as a rider, waiting for the age and stage when she can control the horse independantly and advance in her skills.
I would talk to other riding schools in the area and find out their policies on minimum ages and young riders. Then I would hire a horse knowledgable person to come with me and observe your daughter's lesson and evaluate the safety aspect.
Sorry to be a downer; it's possible that your child *is* the exception and she's in an appropriate safe program, and if so, congrats! But as a concerned parent, I'd have lots of questions.