No, you aren't being narrow-minded (although some people may label you as such). One of the responsibilities God gives us as parents is to guard our children from harm, including moral and spiritual harm.
But our responsibility isn't just to keep them from harm. It's also to help them develop their inner moral character, including their ability to decide what's right and wrong and learn to make wise choices as they grow older. But good character doesn't just happen, nor do children have a natural instinct to do what's right and avoid what's wrong. They have to learn them, and their parents should be their main teachers. The Bible says, "Start children off on the way they should go, and even when they are old they will not turn from it" (Proverbs 22:6).
But what if parents neglect that responsibility, or don't even care? What if they allow the media to become their child's main teacher about right and wrong? We may think it doesn't matter, but it does.
Know what your children watch, and be wise in what you permit them to do. When you have to say "no" to them, explain why. Above all, make Christ and His Word the foundation of your lives. Let His joy fill your home, and encourage your children to give their lives to Christ. And remember the Bible's admonition: "Whatever is true, whatever is noble, whatever is right, whatever is pure... think about such things" (Philippians 4:8).