Exporting graphics for the Web - why do they always look bitmapped??

This has been bugging me for a long time now and I'm ready to reach out to a commuity of people who really know Illustrator. I have been a website designer for many years. But I rarely use Illustrator. I've started using it to create name + image sort of logos to put in the top banner of websites. 

I am frustrated because it seems like I lose so much quality when I export for screens. I really notice a bitmap or fuzzy edge. Is it just me noticing something no one else would? 

I need to use JPG or PNG. These are all Wordpress sites so using SVG isn't an easy option. And I don't want it to be. I just want a JPG or PNG that looks good. I for sure am exporting at a 1x scale. 

Any ideas? I feel like I must just be missing a simple setting. Has anyone else experienced this?

What is the right way to export a logo from Illustrator to use in a space up to 100 pixels high as a JPG or PNG? 

Thanks in advance for anyone's help!!