Torque relationship to speed in a DC motor
Conceptually, you have to think about this slightly differently.  The way I think you are thinking about this is kind of like torque in a vehicle.  A car with more torque is going to accelerate more quickly and is associated with an increase in speed.  In other words, you press on the gas pedal to increase the speed and you need torque to do that.  

However, when you are talking about the relationship between speed and torque of a DC motor, then you have to think about it differently.  For a given motor with a constant input voltage, the speed of the motor is going to be determined by the load on the motor shaft.  For a given load, the only way to increase the speed is to increase the voltage. And this increase in speed will require some more torque to accelerate but after it reaches its new speed, the torque will back off to its original torque (unless, of course, the load is dependent on speed - like in a fan).  

So maybe a better way for you to think about it is instead of saying "Torque and speed in a DC motor are said to be inversely proportional" you say "For a given voltage, torque and speed in a DC motor are said to be inversely proportional."  A speed-torque curve that you see on data sheets is only valid for the rated voltage and the motor will operate on that curve.  So if torque goes up, the speed will follow that curve and go down.