People who are scared of owing more taxes when filing their tax returns are the kind of people who fill out a W-4 form with 0 exemptions.
You can change your exemptions anytime you want to.
You could claim one for yourself. One for your spouse and one for each child and probably still get a refund because of tax credits. If your spouse also works, then you may want to consider splitting that between each of you. If you both claim all exemptions, then you might end up having too much withheld. So, when both parents have incomes, you may want to claim yourself and two children and have the spouse claim themselves and two children.
Basically, when you claim 0, you are giving the government an interest free loan of your money and then you will get it back as a big tax refund when you file.
If you prefer to keep more of your money in your pocket, then change your exemptions. Carefully read the W-4 form and follow the directions for when to claim an exemption. An exemption doesn't always mean someone you are claiming on your taxes. There are other situations that allow exemptions to be claimed. If you are worried about owing taxes instead of getting a refund, then claim one or two less exemptions than what the worksheet says you can claim.