Visualizing the idea

http://upload.wikimedia.org/wikipedia/commons/f/f3/Neornithes.jpg


This chart is far from being correct but it does give the idea of separate branches leading to specific kinds of modern birds. For example visualize the line entitled Ichthyornithiformes continuing into a modern bird line and also Presbyornithidae continuing into a modern bird line - then you are getting closer to visualizing what I am saying.
This would be more complete if it had a line for Hesperornithes (which would lead to another line of modern birds). 
Please note that I am not endorsing this chart in total, but it does present the idea of separate lines leading to lines of modern birds. 


For example, you can see from this diagram that anseriformes and galliformes are branches that run "parallel" to the other lines up to the current time.
Those are two of the lines I am talking about.

You can also see that the diagram includes presbyornithidae and ichthyornithiformes but has them ending at 66 mya. I am proposing that like the other "parallel lines" those lines continued to modern bird lines.

There is more to it than that but this diagram shows how current thinking accepts parallel lines and with almost no change, it is what I am proposing.