Learning the causes and work-abouts of transistor behavior can be a fun past time.
When it comes to speed, you can really slow it down by either driving it with too much current, or by turning the transistor completely off and starving it for current when it attempts to come on.
The most common issue is the one you're seeing. Just like Yungman said, you have extra charge carriers loafing around in the base region. Using you general transistor, you can help with the turn off in these ways:
1 - Turn down the turn-on drive so you don't have as many extra carriers.
2 - Turn your generic transistor into a schottcky transistor by adding a high speed schottky diode.
3 - Pull the extra charge out during turn off with a speed-up cap.
For case 1, take note that there is a comprimise between too little base current and too much. Granted the transistor is saturated when your driving with the collector current. However, unless it's got a really low Hfe, it's probably well saturated at 1/20th the collector current or even less. Some people stick to the rule 10x the minimum you'd need over the range of Hfe. Unless you're preparing it to be neutron hard, I think 5x is good.
For case 2, the schottky diode between collector and base is good, but remember that it needs to be a schottky signal diode. The ones used for power supplies have a HUGE capacitance, especially near zero volts.
Case 3 is my favorite. In this one, you place a series RC in parallel with the driving resistor such that during the instant of turn on, there is excessive current momentarily. Then, the RC allows the drive circuit to yank charge carriers away - thus speeding the turn-off operation.
There is a drawback in that you can actually make the base emitter voltage go negative during the turn off. If it should go too far negative, say below about 4 volts, than the base-emitter junction will begin to behave as a zener. Continual use of the base emitter junction as a zener is okay for use as a zener, but it's detrimental to the long term Hfe of the transistor.
The answer is to place a resistor between the base and emitter that prevents the turn off pulse from going below about -2 volts. This resistor also helps diverting drive current to ground and pulling charge out of the base-emitter during turn off.