More fun with the camera

I've been having a great time with the camera. We went to the playground this afternoon and I took these shots there.

Brienne on swing Laurent climbing Celine at Playground

Perhaps someone can help me... The challenge I'm having is taking clear shots without the flash. If the flash is turned off, which I wanted it to be to get the natural light, then the image is fuzzier. I'm thinking specifically of the photos of Celine & Brienne (I like the movement shown in Laurent's photo since I think it captures how he really is - moving all the time). Any advice for how to deal with this. I'm really ignorant when it comes to photography, trying to learn.

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