Servo motor power boost
The circuit inside a hobby servo is optimized to provide best performance for the size, so all the components are matched to work together. The motor driver output (inside the control IC or external transistors) is sized to suit the installed motor. 

You could replace the transistors with more powerful ones if the IC is able to drive them (perhaps possible with FETs, but probably not bipolar transistors because they would need more current than the IC can provide). To find out if it is possible you should examine the datasheet of the control IC used in your servo. Alternatively you could try tacking a booster amp onto the existing motor output, but this may not work because the controller expects to see a motor there.  

Assuming you can make a more powerful motor driver, the next problem is tuning the servo for fastest response without overshoot. The PID (Proportional Integral Differential) algorithm built into the control IC is tuned to the particular mechanism it was designed for. When you put a larger motor in its response will probably be different, so chances are the servo will work poorly. 

Analog control ICs are usually tuned with passive components (resistors, capacitors) so there's a chance you might be able to tune an analog servo to use a larger motor by changing their values. Digital servos are controlled by firmware inside the controller which cannot be changed, so you are stuck with whatever tuning they have.

In short, while you could modify a hobby servo to use a larger motor, it would be much easier to just buy a larger servo to start with. If you want to take the control board out of a hobby servo and use it to control your own mechanism then get an analog servo with equivalent sized motor, and be prepared for a lot of fiddling to get it working properly.