Let's say we're approaching a headwind @ 30 mph (or maybe 100 knots). Our ground speed, headed in the direction into the headwind is 550 mph. If we turn around, our new 'tailwind' will make us go 650 mph, over the ground.
The question to the pilot: how does airspeed and turning around in a big circle relate to each other?
A: ignore the airspeed as you make the circle or whatever shape. The airspeed needs to be in a range that is critical to flight itself. If the airspeed is too low, the airplane will stop. If the airspeed is too high, the wings will get ripped off (damage to airplane, hairline fractures, damage the airplane).
The key to understanding this is that you measure wind speed, you don't try to counteract it.
Q: When you're up in the air, and you're making turns, if you set the power at a given setting (not going to change the power, try to climb, try descend) then that would change the airspeed field as felt by the airplane. Try to dive, airplane's going to speed up.
If you stay level, and you don't change up/down, not messing with throttle, then the airspeed is going to stay the same. (Airspeed should not to be confused with turbulence).
As you are turning, into or outside of the headwind, your airspeed will not change, unless you do something to change it. The ground speed will change tremendously. Your former headwind will become a tailwind.
The wind will make your idealized circle, as you fly it, end up like a spiral of some sort, or squiggly hair since you can't pull a perfect circle and remember where you were, over the ground, unless you track the airplane from an overhead satellite, and have a Computer driving 'the perfect circle' (i.e. constant adjustments, throttle, angle of bank, etc.)
To date, no civilian aircraft is supposed to fly into a hurricane because there is a lot of turbulence and shear forces that could rip apart a plane. Only specialized military aircraft do this, although the pilots of civilian aircraft are all largely ex-military flyers.