I'm pretty sure those icons are based on the website's "
favicon" and can't be changed by you. But I'm confused looking at your screenshot and doing some looking at my own Safari favorites page. YouTube certainly has a favicon and it shows up for me if I add a bookmark for YouTube, so I have no idea why it's not showing up for you. Same goes for a lot of others you have there with no favicon displayed. Additionally, I know for a fact that my own website has a favicon, and it displays just fine in FireFox or Chrome, but doesn't display in Safari.
Odd. Maybe someone else can shed some light on this.