Under polygons go to tool options and you will see a option for keep faces together. Click on it and now when you do any face extruding the faces will stay together and not turn into what you have happening in you picture.
The reason why the geometry is doing that is the vertices are not mergeded.
Kurt
I am enough of an artist to draw freely upon my imagination, knowledge is limited, imagination encircles the world. (Albert Einstein)
https://www.artstation.com/kurtb