https://www.amazon.com/gp/product/B01KVWIKHG/ref=ppx_yo_dt_b_asin_title_o05_s00?ie=UTF8&th=1

So, I'm really new to owning a Westie (so, take it for what it's worth). We got our 9 week old just 3 days ago. We were asked to give him a bath when we got him so I struggled with what kind of shampoo to get, too. We used the shampoo above, and it's been a few days and we don't see any issues. In fact, he went in for his well-checkup today and passed with flying colors.

I'm sure you're already aware - but you don't bathe Westies very often. I've seen recommendations as little as "every 6 weeks" to "every 3 months". But...nonetheless, I do remember commenting to my wife that he looked much whiter after his bath...but that could have just been "any bath" and not necessarily this shampoo. My main concern was safety and it being gentle on his skin so he didn't dry out.