Okay. Five minutes/mile pace is about 37.3 per 200m. What is your fastest 200? You mention "sub 4 speed" so I'm assuming you can run 200m under 30 seconds--but I shouldn't assume. If your fastest 200 is over 30 seconds, or even 29.x, then yes: sub-5:00 pace is going to feel like something close to a sprint. In this case, training for absolute speed--so that there's more of a "cushion" between your top sprint speed and 5:00 pace--would be helpful.
Beyond that: People have already mentioned training at a variety of speeds, and that's important. If all or nearly all of your training is at 6:00/mile pace and slower, then just the mechanics of running at ~35sec/200 can feel difficult--like you have to shift gears from "normal" to "sprint" without any intermediate sustainable "striding" gear.
We can borrow from Igloi here. He distinguished between two major movement patterns. I think he called them "swing" and "speed," and I think the former corresponds to the kind of economical striding pattern that you want to develop, one that will let you run at a *good* pace for a while but won't fatigue you as rapidly as a high-knees sprinting-type action.
So go with another part of his training: lots of short (or "shortish") repetitions at that intermediate speed, with striding (not sprinting) mechanics. His guys were always completely comfortable and efficient at their mile goal pace because they had run it hundreds of times in practice. (Whether they had the stamina to *sustain* that goal pace was a separate question.)
Keep it simple. Once or twice a week, warm up and then do sets of short repeats, with moderate rests between reps and good rests (maybe jogging) between sets. Focus on a relaxed "condensed" stride with relaxation in the elbows (and shoulders and neck and face), getting your speed more from turnover than from stride length.
DON'T GO TOO FAST, which would tend to push you into sprinting mechanics. Look for something like 17-18sec for 100m or 35-36sec for 200. I actually recommend 150m segments (in roughly 26-27sec), because people don't have as much "invested" in hitting fast times for that distance. (Some people enjoy running 100m on a soccer pitch or 110yds on a football field, esp. if the surface is grass. A natural surface promotes a more compact, efficient stride.) Whatever distance you choose, you can start with sets of, say, three or four sets of 3-6 repetitions and build to sets of 10-15 reps.
DON'T GET TOO TIRED. Make sure that you're not developing a big oxygen debt, because that would push you into flailing or fighting your way through the later reps and defeat your purpose of building a smooth efficient stride pattern. As time goes by you might start using longer repetitions (250 or 300 or 400 or 500 meters) in some of your sessions, but only if you have the efficiency and relaxation to run the same pace over those longer distances without building a big debt.
DON'T DO THIS AFTER A DISTANCE RUN, because when you're learning new motor skills you need to be fresh. Get an adequate warmup, then move into your repetitions. You can *finish* the session with some easy mileage, if you wish.
I hope this can be some help.