KevCol: I agree that the rule of thirds can add artistry to a photo, but more important is framing, which I think could be much improved on a few of these pics.
I like to see the subject offset provide the space for the subject to move into or to be able to look into. A frame edge can seem like a wall if placed wrongly and the half a horse and the girl look confined by the frame.
Also, tight cropping can look uncomfortable if it dismembers objects even to the extent I would not have trimmed the kids shoes or toy. This shot would look less bland and have more of a narrative if there was a hint the child actually had been able to play with the cart rather than it had been poked in a corner by a tidy adult.
Maybe his surrounds could have been improved by having him sitting on a rug and even have a cushion to cover some of that skirting.
I should say your child looks cute and I like the hint of natural lighting.
I think the girl picture could be much improved if the photographer shot from the other side of the fence (producing almost a mirror-image of the example). The subject will always look a bit confined if the direction they are looking at is cut off abruptly by the frame.
But I think "dismembering" subjects in and of itself doesn't always amount to an uncomfortable composition, on the contrary, very tight portraits for example usually work well when you cut off the top of the head and chin (so that the eyes are on roughly the top "third" line).
An even more obvious example is a full-length portrait. Including people's feet makes the picture awkward, but if you cut them off say, mid-calf, it looks much better. Try it!
steveh0607: A good article for the basics. But sometimes a charging horse looks more dramatic when it fills the frame and is coming straight for you.
I would say with the charging horse one, it may look more natural to frame it so that the horses look like they are entering the frame rather than leaving (or in this case looking like they have already left) the frame.
Taking into account that most folks subconsciously "read" from left to right should also help in making a framing decision - good composition leads your eyes into the picture, not out of it.
An alternative way to indicate motion is to play with shutter speed/aperture settings. Either blurring the background or blurring the horses can work.