It can be difficult when you're excited about an idea, and your family's reaction isn't what you'd hoped it would be. First thing to remember is that they mean well. Both your first respondent (Keith Sun) and I are trained counselors, and I can tell you that much of our coursework is dedicated to the seemingly simple act of responding appropriately to what people say. We have this hardwired instinct to offer advice and input, even when the most effective thing for us to do would be simply to listen and affirm what someone's saying. I think that it probably comes out of concern.
While you were presumably hoping that your family were as excited about your idea as you were, think about it from their perspective for a moment. But remember, I'm not telling you to accept their perspective. Just think about it for a minute. They've been where you are, in many cases. And they've been on that long stretch that comes AFTER where you are. So they're thinking about the long game. What are you going to do with this major after you graduate? What are your job prospects? And so on.
As an academic counselor, I used to tell students that families aren't afraid of you not being an engineer or lawyer. They're afraid of the unknown. And if this major is an unknown to you as well, then step one should be research. If you go to them and say "I'm interested in fashion merchandising; the courses look really interesting," then that doesn't allay their concerns. And, believe it or no, somewhere along the line, you'll have the same concerns yourself. What the people in your life want to hear about is a plan, I imagine. So my advice is to do your research and put together the whole picture for them. And for yourself, ultimately. Whether you ever convince them or no, you'll feel a lot more confident in your decision if you've done the homework to figure out the plan for yourself.
What does that mean? Start doing some research on a site like O*NET, which shows information like pay ranges, education levels, hiring trends, required skills, and so on for various careers. Do research on the sorts of places you could do an internship, what exactly you'd be doing at each step of the way, and so on. Paint a full picture, so you know what you're getting into and can fully explain it to your family.
Perhaps that process will solidify the decision in your own head. Perhaps it will convince your family. Perhaps you'll be dissuaded and select something else, but you'll do it from a position of knowledge, not just self-doubt.
Don't worry. What you're doing here is perfectly normal. And what your family is doing is normal as well. In fact, I'd say that this sort of self-exploration is a large part of the actual purpose of higher education. There's a reason why colleges have career centers, advisors, counselors, internship programs, etc. Colleges and universities are like a lab. Now you need to do your research and run your experiments. You'll figure it out.