It sounds like you want a masonry type of layout. It's not too difficult. You would first need to modify your postformat.php file to remove the post excerpt (i.e., the text description), and then rearrange the elements so your post title is underneath the thumbnail. The recent posts in the widget don't show a post date or author name, so I don't know if you want to keep them or leave them in.
Then you'll want to add some CSS that formats the posts across four columns (it looks like you'll have room for four in a row). You probably need to resize the font size for the post titles so they will fit in a narrower column. You'll also want to add some media queries to change the number of posts to two across when at a tablet width, and one post across at cell phone width.
I created a sample page
here that shows how to arrange posts in two columns. The principles are the same, you just need to make adjustments in the CSS for four columns.