I'm buying a bunch of these babies below to create a privacy screen in my backyard. I'm getting young ones and want them to grow as fast as I can make them grow. Is there anything I can do to make them grow extraordinarily fast?

I'm buying a bunch of these babies below to create a privacy screen in my backyard. I'm getting young ones and want them to grow as fast as I can make them grow. Is there anything I can do to make them grow extraordinarily fast?

The time you spend in preparation is as, or more, important than post planting time.
Post planting
What you don't need to do
There's an interesting piece of research online carried out in June 2005 at the Connecticut Agricultural Experiment Station, and published in the Horticultural Research Institute Journal; one of the subjects in the trial was Thuja Emerald Green. Under control conditions, they were planted with various amendments (bio gels, mycorrhizal fungi and various other products) with a control which had nothing added other than 3 inches of pine mulch at first planting, and a year later. Most were finger weeded in the immediate area during the two year trial period where necessary.
The results show that the Thuja planted with nothing but the addition of a 3 inch layer of pine mulch on top of the soil did better than all the others - the mulch was applied at planting time, and then again a year later. The conclusion from the trial was that this worked best because the mulch retained moisture and kept down weed competition to the plants, though there is a caveat - it depends on the state of your soil currently as to whether they would benefit from other treatments. If your soil hasn't recently been brought in from elsewhere, or been graded or disturbed greatly, and it isn't particularly prone to drying out completely, then planting with the 3 inch mulch of pine would seem to be a good choice. If your soil tends to dry out rapidly, the addition of Soilmoist Gel into the planting hole, around the roots, will make a difference, but the mulch should still be applied.
Otherwise, Kevinsky's answer gets a vote from me.
Thujas like a fair amount of "food" to jump start them after xfer/transplant. Purchase some sort of evergreen fertilizer product ("M Roots" is one name brand) that has a mycorrhiza element within it as these are the fungi that help with the nutrient transfer for a healthy plant. Also, we have used a 1:1:1 mixture of organic compost, loam, and peat moss for the process of back filling the root ball in the hole. Avoid "dirt" with a high rocky content...if the original dirt from the hole is good, you can use that with the compost and peat... but avoid clay heavy or rock heavy soil. Sandy soil is ok though...but add a bit less peat and a bit more loam if that is the case.
Depending on your space...STAGGER your plants making an equilateral triangle between 3 plants. 2' distance on each side or more (how high did you want to make this hedge?). 2'is great for a 6-8' hedge. Leave more distance for a taller hedge. If you see one of your arborvitaes begin to 'flag', a particular branch versus all over, get it diagnosed (Cooperative Extension Service). It might be a very virulent disease and you'll want to get it out of there...soil and all. But if you lose one or two having planted in a staggered row, it won't be as noticeable if at all.
Make sure that you do not dig the hole any deeper than the height of the root ball. Trees and shrubs, woody plants, have a very distinctive line between ROOTS and STEM. Only the roots get covered with soil, the woody stem cannot be covered with soil, mulch, paper, rocks...this allows bacteria needing moisture to start decomposing the bark/stem. Very shortly this will effectively girdle your tree or shrub, a year or two. I even plant leaving an inch or two of the rootball out of the soil so that I can add mulch.
I once used Thuja plicata 'Virescens' for a screen. I made a 'hedge' in a more natural way, grouping 3-4 at 5' oc making a break (a small grove of elderberry with dark purple leaves) then adding 2 more Virescens Cedars (thick, fast-growing, narrowly pyramidal tree), one staggered front, the other rear, another matching elderberry and then a grouping of 3 Virescens. Just an example...within 2 years we had a dependable, low-maintenance privacy screen. In 9 more years, we still did not have to prune the tops of the trees and we had an oasis. Great for the wind, too. At the time, the Virescens cedar was less expensive than the arborvitae for the same height.