Overall I would say the new browser UI is an improvement over the previous one. The filtering of bookmarks while searching is really nice, as well as the improved history view and the new options in the settings. The browser engine finally being aware of the keyboard and making sure that elements anchored to the bottom of the viewport is also a great addition! The landscape layout in the tabs overview page is also neat.
The popup menu is way better than the enormous expanding toolbar in previous versions in my opinion. I believe the discoverability issue of the extra menu items could be simply fixed by showing 6.5 menu items by default instead of 7. It may not look perfect, but I’d say it’s an easy, practical solution.
I would suggest in the tab overview screen to remove the back and the new tab buttons. These are already redundant by the back swipe and pulley menu. I would also suggest to either move the private browsing buttons to the bottom or to put it back in the pulley menu, as they are impossible to use with one hand on the Xperia 10 II. Lastly I would consider to increase the size of the close buttons. I feel like they are a bit to small to reliably hit.