Electronic – S8050 transistor currents not like in the datasheet

The typical gain of a 2N3904 is 200-300 at a couple mA collector current (more as it warms up due to not being saturated)

Even with 560K, that's half a mA or so, which will give plenty of light from a modern LED, but you should be able to see that it's not as bright as when a 10K resistor is used.

Do not use the hfe for this calculation if you want the transistor saturated hard on, use a forced beta of something like 20 to 50, if the typical hfe is 200 or so and the minimum 100 or so. If you use, say 30, in your equation you get a resistor value of 8.8K, so you might use 10K or 8.2K.

The reason is that you won't likely have a guaranteed hfe for the current you're using, and the hfe decreases at temperature extremes. It's still only "wasting" a few percent of the LED current, so no big deal.

To prove this to yourself, take a voltmeter and measure Vce of the transistor when it is on. If it is something like 50-100mV it is saturated.

There are two possible causes of the blown transistors. First, connecting the base directly to 3.7V will cause excessive base current : it makes sense to limit that with a resistor. 20mA is cautious, 100 ohms would allow 30ma which should be safe.

The second in that - while you cannot "push" excessive current like 2A to the motor - if it draws 600mA when running normally, it may draw much higher currents when starting, or when stalled.

Connect the motor to the PSU again, and stall it with your fingers (don't do this with a high power motor! Use some other means of braking it!) Observe the current it draws when stalled ... I'm guessing somewhere over 3 Amps (you may need to fiddle with the current limit to observe this).

It will momentarily draw that current when starting, and that could be fatal to a 650mA rated transistor...