Worry is one of the jobs of parenting. Stuff that you never thought about for yourself--how often you poop, whether you should eat honey or not, how many inches you've grown in the last six months, whether your teacher likes you or not--becomes of paramount importance when it's about your kid.
That's all normal. I think it's biologically wired--if early people didn't worry and keep their infants close, those infants would be stolen by dingoes. We're still human, so it's still hardcoded in us to keep kids close and to worry about them. Thinking through the possibilities and how we'd deal with them helps us with mental flexibility and keeps us prepared for the inevitable crisis situations.
If your worry becomes so big that it takes over other parts of your life, and prevents you from having other emotions about your child and the other things you do, that's a sign that your hormones are out of whack and you need help. Tell your partner or a friend and they'll help you tell your doctor, and your doctor will get it straightened out. Overwhelming worry is treatable.
But normal worry, worry that's just one occasional emotion mixed in with all the other emotions, is part of being a parent. As your kids get older and more competent your worries will grow with them. But you'll be able to meet each stage, prepared.