So, you’ve discovered you’re not the true owner of your website. You pay your webmaster for the registration as well as the hosting, and you paid for the design of the site. Taking all that into account, how can you not be the owner???
When you retained a company or individual to build your site, the first thing they would have done was to register a domain name on your behalf. When they did that, apparently they listed themselves as the Registrant (or owner).
Let’s give your designer the benefit of the doubt and presume this is not because they want to bully you into staying with them as your provider. Let’s say it’s a matter of convenience or because they simply don’t know any better. How do you remedy the situation?
Well, if you’re on good terms with your designer, you could simply ask them to list you as the Registrant. If the relationship has become less than amicable, however, you will need to initiate a dispute through a Dispute Resolution Provider like the National Arbitration Forum. If their determination is not in your favor, you can escalate your dispute through the dispute resolution center of the Internet Corporation for Assigned Names and Numbers (icann.org).
When all else fails, you may need to find an attorney who specializes in Internet law. I certainly hope it doesn’t come to that for you and you’re able to reach a resolution before it escalates to that!