A 5 y.o. kid sometimes treats the process of hair washing as an invasion into his private space. Due to this factor a conflict appears between mother and kid, a mom needs to wash a baby hair, and a baby does not allow her doing this, as a result each trip to a bathroom is accompanied by tears and crying.

How to make a washing procedure more pleasant and painful for a mother and kid:

Find more time for hair wash, this way you won’t have to hurry up. A kid also should be in a good and calm mood: not hungry, tired or upset.

Take a doll whose hair a kid can wash too in a bathroom.

Use shampoos “without tears”.

If a kid does not like water coming into his ears, use earplugs.

To make a procedure more interesting, build different figures of hair and foam. A cap’s peak or underwater spectacles will prevent water from coming into a kid’s eyes.
Give a cup to a kid and he will try washing hair by himself.

These are just several variants you can use. Probably, they will help you inventing something new. If nothing helps, do not force events, just try washing hair only with water.