Umm.... in the title, LT said the cultivar is Twisty Baby.
You are right, LT, that heavy pruning (and extra water or fertilizer) can stimulate faster straight growth that is abnormal for Twisty Baby. Your goal should be to promote normal "twisty" growth by not treating it with extra care, as you might a perennial to get more flowers. As far as cultural care, just let it be.
That the growth is horizontal is not concerning, but I would want to conserve the somewhat contorted growth pattern that is normal for the tree. If it were me, I would first prune out one-third of the straight growths completely. Not 1/3 of each one, but 5 of the 15, completely. In another month or so, take 5 more off. Next year, take off the remaining 5. Hopefully, this slow-deliberate pruning will not encourage the uncharacteristic straight growth of the cultivar. Be sure to cut all the way back to where there seems to be the normal "twisty" growth that the cultivar is known for. If you only cut back a straight branch by half, the new growth that ensues will like be straight as well.
