Coloring is a skills that helps develop hand strength, hand endurance, visual attention and eye-hand coordination.
Coloring starts with marking and then develops into scribbling. Coloring skills develop down the arm and control of the crayon becomes more refined.. Control begins at the shoulder, then elbow, wrist and finally to the fingers. The smaller the area you are coloring the more refined control you need.
Use broken crayons to help your child use a fingered grasp. Jumbo crayons or knob crayons are great as well!
Start with a small 3-4" picture with simple lines for a 3-4 year old. Give them a broken crayon to hold to strengthen their grasp. Have them imitate coloring with vertical strokes. Then try horizontal strokes. Three year olds are not always able to hold the paper with their "helper hand" so try taping the paper down. Four year olds may not remember to hold onto the paper, so try putting a post it note on the side to have them cover with their hand.
Also try coloring on a vertical surface like the bathtub wall or wall with paper if your child understands it is not ok to color on the wall. This helps develop the wrist muscles and strengthen grasp for coloring with better control.
Also try using a texture for the border. Use glue to trace the lines and let it dry. Now it will be bumpy when you child colors over the line.
Vertical surface coloring
Scratch art paper or textured surface
Find a texture they like to color on top of like a rubbing plate, or leaves from outside. This also helps with crayon pressure.
You can also try using scratch art paper with a stick to rub off the top layer.