Turbinado Sugar is made by crushing the sugar cane, crystalizing the juices and drying the sugar by spinning it in a turbine. The word turbine is where the word turbinado comes from.
Demerara sugar first came from Demerara, Guyana. More than anything else, it is known for its crunchiness and brown colour. It is typically used for baking and as a method of sweetening coffee. It is also used to darken light desserts as effectively as turbinado sugar.
Light Brown Sugar
Brown sugar is simply white sugar that has been coloured with molasses. This type of sugar is very heavily processed. Light brown sugar is dyed with a lower amount of molasses than dark brown sugar, and brown sugar in general is extremely sweet.
Raw sugar is not actually raw. The sugar cane is pressed, and the resulting liquid is combined with lime. This changes the pH balance. Then the sugar is evaporated and the crystals separated into "raw" sugar. The reason it is called raw is because there is still some molasses that has not been removed from the sugar.
As these are all made from the same sugars, use the alternatives in the same measurement as you would use turbinado sugar. Some other forms of sweeteners may require different measurements however.