I use these from Amazon EWIN(R) 4pcs Soft Adjustable Baby Kids Children Shampoo Bath Bathing Shower Cap Hat Wash Hair Shield Hat https://www.amazon.com/dp/B00LHPV5M4/ref=cm_sw_r_apan_glt_i_WH9EQJT5EPFJW5R6KJ78?_encoding=UTF8&psc=1 . I let my daughter choose if I wash her hair while she takes a bath or shower.
Have you considered using something like a shower visor. I used something like that when I was a kid and it worked well.