Yes indeed a 3 year old does experience small growth spurts and they are growing everyday that passes even though you may not notice it.
You usually don't notice that a child has grown until at least 6 months to a year has passed and then you see them getting taller and growing out of their clothes.
However when the child is a newborn you're gonna notice they are growing fast and within a month they will usually need bigger clothes and bigger size diapers as well.
Major growth spurts happen between the ages of 8 and 15 years old where you'll notice major changes in the child's growth.
The child will usually be more hungry and want to eat more than they used too as well.
You should also notice that as a child reaches 3 years old they will usually not have that rounded tummy like they did when they were 2 years old and younger.