Yes cutting grass does make it grow faster and healthier than not cutting the grass.
When you cut the grass you cut off the tip of the grass that contains hormones that are released from the grass when help to make the grass grow faster, thicker and healthier.
So if you cut your grass before it gets too long then it will grow back faster, thicker and healthier.
That is as long as you do not cut the grass too short or scalp the grass.
Make sure your grass is cut to a good height and your lawn mower is not scalping the grass.
Also make sure your mower blade or mower blades are sharp and ready to cut the grass properly.
Dull mower blades may cut grass but it doesn't do a good job at cutting grass.