Turbinado sugar is a naturally brown sugar. It is natural because it is only partially processed sugar cane so some of the full bodied flavor profile of the cane juice is left in the sugar. This is opposed to brown sugar where the sugar is refined to a greater degree and then molasses is added to the sugar to give it it's distinctive color and flavor.
So with turbinado sugar, the full bodied flavors never leave the sugar in the first place. Turbinado sugar is processed somewhat and the name refers to the turbines that are used to spin the cane extract to remove some of the heavier compounds. But turbinado sugar retains much of the character of raw sugar cane juice.
In a lot of drink recipes, turbinado sugar can be substituted for white sugar to give a drink more body and character. In the same way a syrup made from turbinado sugar (rich syrup) can do the same thing versus a simple syrup.
When substituting turbinado sugar for white sugar or superfine sugar keep in mind that turbinado crystals are quite large and take a lot longer to dissolve. Your mixing time for your drink recipe will go up significantly. One solution is to mix the turbinado sugar with a little water and agitate this mixture until the sugar dissolves, then continue with the recipe. The easiest solution is usually to just use a turbinado sugar based syrup such as rich syrup or rich gum syrup.
Those large crystals do a great job when muddling with citrus fruit. The big crystals scratch and gouge the lemon, lime, or orange peel releasing oil in prodigious amounts. But you still have to problem with dissolving the sugar.
There is a sugar that is almost identical to turbinado sugar called demerara sugar. It is named after a river in Guyana and is the common name for this type of sugar in Europe. Demerara sugar can be work as a turbinado sugar substitute using the same quantities.
Another substitute is called natural or raw sugar. It is lighter than turbinado sugar but will do the trick in a pinch.
Brown sugar whether light or dark can also be used but with the added molasses you will get a different flavor profile in the finished drink. If this is all you have choose light brown sugar.
Switch out your white sugar for turbinado in a drink recipe and let us know how it turns out.