How to teach a kid geometry
Solution 1:
I've assisted in teaching kids aged 10–14 mathematical art, and found a lot of success with the following project:
Invent a polyhedron and try to make it! Basically, take construction paper, and have them use straightedge and compass to make shapes and subsequently cut them out. The most "appealing" are regular shapes. Since your child is 4, you should probably cut the shapes for them.
Here were some especially nice moments:
One student tries to make a vertex with $6$ equilateral triangles and asks me why it wasn't working.
Multiple students worked on constructing pentagons, which was a bit of a challenge (for me too!)
Some students finished early and started coloring their polyhedra, with $4$ or $5$ colors. Of course I bothered them by asking if they could have done it with fewer.
Some students couldn't "See" the shape, so we would use projections to form planar graphs, or use "nets" to help. This was often a bit of a brain cramp but a lot of fun.
here is a blog post chronicling a collection of activities we did (some were more advanced than others: triangle inequalities, pythagorean theorem, finding a way to approximate pi with arbitrary precision etc.) that might have some nice ideas for you!
Solution 2:
If it were me, (assuming a child in elementary school) I would start heading into drawing with a ruler and compass.
Working on accurately measuring lengths and angles, drawing circles, lines, and perhaps other shapes, depending on the tools you add, would introduce a visual-art aspect that might help fuel further interest.
For instance, you could work on constructing an equilateral triangle, then adding a few iterations to make it a Sierpinski triangle.
Or you could construct the various centers on a triangle and show they're collinear.
Or you can work on drawing some simple fractal patterns.
Color and pattern goes hand in hand with art and mathematics.
Origami is also a good way to go, and is a good option for children who can't be trusted with sharp implements. There's a book called "Geometric exercises in paper folding" by T. Sundara Row that was passable, but I imagine there's newer and better books like that out now.