Firstly, anyone who knows me around here knows my stance on all things feminist. I don't buy into the sexualisation of anyone. I stand up for individuality. Kids should be free to be who they are, regardless. I'm a support of equality in all forms. That everyone is perfect who they are no matter what.
Having said that. It's hair. It's one part of you that you can do whatever you like, and eventually it grows out/can be cut off/grow back. So, yes. I think it's perfectly okay to let your 4yo dye their hair *depending on the reasoning*.
If your kid is all like "hey, I hate my hair, I hate the colour, I'm being teased about it at school, it's awful" etc etc then no, you're not going to let your kid dye their hair, because the underlying issue is with self confidence, bullying and body image issues. Allowing them to dye their hair unquestionably in that situation is not the right thing to do and it sends the message that it's okay to change who you are for those reasons.
However, if your kid comes to you and is like "yo, I reckon it would be AWESOME if I had red hair. Can I have red hair? That would be AWESOME! Pleeease It would be sooooooo COOL!" then I'd be like, well, y'know kid, it doesn't wash out. You'll have to have red hair for weeks, maybe even months until it washes out or grows out or you put another dye on it. Dying your hair also could mean it's not as soft and lovely anymore because it's not good for you hair. If they continued to pester, then sure as heck I'd dye their hair if it made them happy because they thought it would be awesome
Srsly. People pierce their BABIES EARS, dying a 4 year old's hair has nothing on that, despite the fact that it's less socially acceptable. That's it. It's socially acceptable to let your child have pierced ears, but not to dye their hair. I think that's stupid. So yes, if it was simply a cosmetic fun thing, and not one of the reasons I first listed, then yes. I'd dye my kids hair