How do I export an SVG from Illustrator such that it renders well in small sizes?

I come from a Photoshop background and this is my first time dabbling with AI.

I would like to show my product logo in a small size on a webpage (about 15px height). I currently have a PNG (created in Photoshop) but the font is quite choppy. I am assuming it has to do with the scale.

To fix it, I am now trying to work in Illustrator and saving as SVG. I've tried a bunch of things but the logo shows up very nicely in large sizes, but not small.

Here's how it is big:

But here's how it shows when it's close to the size I am looking for:

How do I make the smaller text smoother on my webpage?

I have tried setting anti-alias in the Character window but it's disabled. Should I even try using an SVG or should I try fixing my PNG in Photoshop?