It’s theoretically possible to fill most of the header, but will probably take some trial and error to get it right. You may find that you have to adjust your site’s CSS in order to do it. If you are interested in that, have a look at Beginner’s guide to using Discourse Themes and Developer’s guide to Discourse Themes.
Looking at our instructions for the logo setting, I think we should give an optimal aspect ratio. The logo in your screenshot does have an aspect ratio of greater than 3:1, but it seems to me that it is too wide of a logo. If you look at the logo that’s used on the Discourse Meta forum, you’ll see that it is displayed at approximately 150 x 40px. That seems like the ideal aspect ratio to me - slightly less than 4:1.