I am trying to join a 2x4 supporting a workbench tabletop to another to make a backsplash of sorts. Any ideas how to do this without dowels? (No, I can't use 2x8)
I would definitely use Pocket-hole joinery
My brother got me a jig that is very easy to use, and the joints are incredibly strong even without wood glue; also if you use glue the joins are "self clamping".
Basically this allows you to edge-join, or do 45 degree angles, etc. just by drilling a couple of perfectly angled holes and screwing the pieces together such that the screw doesn't protrude from your work.
Obviously, I wouldn't get a jig for just one project, but I have totally stopped using dowels, biscuits, and other fancy techniques in favor of pocket holes.
Hope this helps!
You could use some form of strapping on the back side.
With this option, you can extend the strapping and use it to attach the whole bit to the workbench.
You could bore holes on the edge of the top or bottom piece to about half the width of the board, then use screws to fasten the two boards together.
You'll likely want to use a drill press to bore the holes.
Since it's not going to be structural, you could simply glue the two boards together using wood glue.
If you want to get real fancy, a sliding dovetail would work.
Though with this joint, you'll loose a bit of overall height.
As you say you want to use screws and it's non-structural, I'd go with:
Get some 4" wood screws, drill half way down through the upper piece with a drill slightly bigger than the screw heads, you shouldn't need to do pilot holes all the way through as 2" really shouldn't split unless you use huge screws.
I'd go with 4 or 5 screws along the length.
Screw down through the top piece until you get a little bit of the tip poking out (say 1mm)
Align the top piece on the bottom piece, give it a knock to set the exposed tips into the lower piece, and screw down.
If you don't want exposed holes do it the other way up, if the lower piece is not already set in place.