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Did you happen to use Flowed text? Did you click and drag out a box before you started typing? Flowed text has some problems on the internet. Actually I'm not sure what the problem is, or looks like. But if you did use Flowed text, instead you should use Regular text. Just click and start typing.
However, if that's not it, we'll need to look elsewhere. Another problem could be that the font you used in the image isn't installed on the computer that you're using to view the webpage. When your browser opens a webpage that uses some particular font, and you don't happen to have that font installed, there's no way for your browser to display it. So your browser substitutes a font that it does have access to.
In order to avoid this problem, you can convert the text to path. As a path, the browser won't need access to any font, and will display it just as in your image. Path menu > Object to Path to convert to path. However, after you've changed the text to path, it won't be editable as text. If you think you may later need to edit the text, you could save the text as text on a hidden layer in the file.
I don't think a font can be installed on a website....at least not in a way that will allow any computer visiting the webpage to view the text using that font. Although I could be wrong about that
