Like you, I had the same arguments about appearance with both my son and daughter when they were teenagers. You asked me what I did about these issues. I want to let you know, one parent to another, it isn't as awful as you think.
When my son was 17, he told me that he wanted to get a tattoo and some piercings. Instead of arguing, we talked about why he wanted them and also about the future consequences. I helped him to think. It took a while, but he finally understood that tattoos are forever and decided to wait until he was older. But I agreed to let him get a small ear piercing. So, we compromised.
One morning, when my daughter was 16, she was leaving for school wearing a black leather dress, tights, and high black boots. She'd dyed her hair pink the night before. I was shocked. I yelled, “You are not leaving this house dressed that way, young lady!” I remember the exact words she said to me. She very calmly told me, "I'm sick of you trying to dress me in the clothes you want, to cut my hair like you like it, to make me act like you...I am not you! I'm my own person, and you have to let me express myself in my own way."
At first I was so angry I couldn't speak. But then I quickly realized that she was right. I gave her a hug, smiled, and told her to have a good day at school. After she left, I sat down and thought about all the quarrels I'd had with my parents when I was her age. Most of them were about hair styles, cosmetics, and clothing. I wore some horrible fashions and did some ridiculous things. But I was trying to become an individual. And because of that teenage phase, I became a well-adjusted person. For the same reasons, our children will also grow into wonderful adults.