Seiran
This might make things a bit clearer as to the design compromises involved:
To a first approximation, propellers are a constant 'power' device (it doesn't much matter the source of the power - piston, turbo, etc.). Thrust = Power/Velocity, so the faster you go, the less thrust a prop produces. That means great takeoff performance, but lousy high speed performance.
Again, to a first approximation, turbojets are a constant 'thrust' device - so using the same equation, the faster a jet goes, the more power it produces. Lousy low speed/takeoff performance, great high speed performance.
There is a third relationship - fuel efficiency. Jets accelerate a small amount of air a large amount, while props accelerate a large amount of air a small amount. Maximum efficiency comes when you accelerate an infinite amount of air an infinitesimally small amount - meaning props are more efficient.
Turbofans combine the best of both worlds - they achieve much of the good low speed performance and fuel efficiency of props, while losing relatively little thrust with increasing forward speed so power output increases with speed.
Highly simplified, but that's the basic concept.