Why is thrust available constant with speed for turbojet engines, when it varies with speed for turboprop engines?
Turboprops and turbojets - or, more broadly, jets - produce thrust in somewhat different ways.

First of all, let's address the way thrust is produced. Per Newton's 2nd and 3rd laws, force equals acceleration times mass, and an action (accelerating the air) produces an opposite reaction. After canceling out the variables (the math is easy to find), thrust is proportional to T=v*m' (m'=mass flow rate), and power transferred to the air is proportional to P=v^2*m'/2. All velocities are  in the airplane's frame of reference.

Now let's go to how engines produce this thrust. A jet engine first decelerates the incoming air to a near-zero velocity, generating drag, then accelerates it to a constant velocity, higher than the initial one, producing thrust. Both v and m' for a jet engine vary across the envelope, but they change much slower than the plane's speed. The engine spends roughly the same amount of power per unit thrust at any velocity.

A propeller doesn't decelerate the air at all. It only accelerates the incoming air by some amount. This converts power P into velocity v. Now, per above, P~v^2. If you encounter air moving at 0 m/s, you need 5 kJ/kg to accelerate it to 100 m/s. If the air is already moving at 100 m/s, you need 15 kJ/kg to get it to 200 m/s.

The result is that propellers encounter less resistance at low airspeed, so they get more thrust per horsepower the slower the plane is. This allows them to push more air or push it faster, producing more thrust. For fixed-pitch propellers, this works by producing more air delta-V and less drag. High-performance turboprops tend to have variable pitch propellers, which will be adjusted to push more air (slower) at low airspeed, or less (but faster) at high airspeed.

Turbofans, combining a fixed-pitch ducted propeller with a turbojet core, are somewhat in between. They lose some efficiency and some thrust as they gain more speed, like propellers, but their curve is much smoother and closer to turbojets in this regard.

The above is an extreme simplification (as is the jet's thrust curve in the question), just enough to get the idea across. The jet's actual thrust is also non-linear, which has been addressed in another question: How (and why) does engine thrust changes with airspeed?