If I had that bush....I'd try braiding some of those wiry branches together and form a thicker, composite, main stem, and I'd also stake it. Cut away....better yet, root all the extra branches. The braided stems will eventually fuse together and form a thicker trunk. It will take a few seasons, and you will have branches towards the top of the stem. Of course, continue to braid, and stake, as the individual branches elongate, and stop at a desired height. This training method, using braiding, is done with Ficus benjamina and Hibiscus, to create "patio trees". No matter what you finally decide to do with this multi-stem bush, it will take a few years to get it under control.
Just a few thoughts.... as an "unconventional" alternative.
Frank
(I wonder if your Brown Turkey started life as a tissue-cultured plant)?