What can the race teams in F1 do to lessen the amount of turbulent air behind the car to make passing easier but still have enough downforce on the rear of the car?

What the F1 car designers are doing is to take large trailing edges and put a "taper" on them so they "taper down" as you move along to the back of the car. Note that air moving through the coolant heat exchangers (radiators) and the engine exhaust is just "dumped" out the back of the vehicle, so some air is effectively moving "through" the vehicle. This sets up a condition where the air through which the F1 racer is passing doesn't have to all "spill around" or be "ducted around" behind the car to "fill in" the space behind it. Between the tapering of trailing edges and the ducting of cooling air and exhaust into the space behind the car, drag and turbulence are reduced.

Note that the new cars apply enough ground effects fairing ("spoilers" or "wings") to permit the car to literally fly through the air if its thrust could be used to spin a propeller or a turbofan. But the air behind the car is carefully redirected here and there to insure there isn't a lot of extra turbulence behind the car. The wheels are basically left out in the open, or at least largely so. Use the links below to look at some "ideas" regarding design, including a brief "history lesson" on the vehicles.

The cars are designed as low as possible, and are quite wide for increased stability. To cite one example, there is a design consideration for these machines that is "translated" over from that of aircraft: cross sectional area. A brick with a sharp nose and tail has a given cross sectional area in the "long" direction. The same brick turned sideways and given a sharp nose and tail has a broader cross sectional area, and (all things being equal) will require more power to move it through the air. But a cross wind will have a more negative effect on the longer shape than the shorter one. See how the problem continues to compound itself? Want less cross sectional area? Make the car narrower. But you give up stability. The crux of the matter is that design is all about negotiation.

The "mystery" of designing an F1 car is not impossible to see. By looking at some images, listing and assessing the variables, and then thinking the problem through, you can get a pretty good idea of the difficulties the designers face. It isn't that you'll be able to compete with them, but by looking at some images and thinking about the problems, you can get an idea of what's in play. Fairing adds weight to the car, and ducting air costs in performance as it increases drag. What's a poor F1 designer to do? Let's look at an example of a trade off.

Consider that the carbon fiber rotors of the rear disc brakes of the F1's are out on the wheels. If they were built inboard alongside the transaxle, they could be "sprung" and not left unsprung with the wheels and tires. The body of the car is "sprung" or suspended, and the things "on" the wheels and tires and on the end of the A-arms are unsprung, or unsuspended. Engineers know that by reducing the unsprung weight, the car will handle better, but the rotors cool better and with a lot less drastic air ducting when we just hang them on the end of the axles with the rear wheels. They add to the unsprung weight, but it's a trade the engineers are willing to make.

Links can be found below to images so you can look and start thinking. There is a bit of a "tutorial" in viewing the links in the succession suggested. In closing, downforce in the rear is a fairly independent problem from the one of turbulence behind the car. There is a number of of things that a designer can do with the wing in the back, and that doesn't have much to do with turbulence behind the car (tho' there is a some turbulence associated with the wing, to be sure). And why would an F1 car builder care about making it easy for the car behind to pass? Less turbulance behind a car will give advantage to a car that is drafting and looking for a chance to pass. But a less turbulent design helps the first car, too.