Energy is average power times time: \$E = P \times t\$. Energy is conserved. Since power is just instant moments of energy, it is also a fact that, in general, power is also conserved. So you will often see statements like "power in must be equal to or greater than power out." On average and on human scales of time and locality, that's pretty much also true. But it is the conservation of energy that really rules the universe as we understand it.
Power, itself, though is like a kind of coinage. Each coin has two faces: voltage and current. Power is the product of volts times current: \$P = V \times I\$. Note that neither of these are time. So there is no implication here that current (\$I\$) must be conserved, because voltage (\$V\$) can be adjusted. There is also no implication that voltage is conserved, because current can be adjusted. On average what is conserved is energy and power. Not voltage and current.
The energy per unit time (power) comes from the power supply. That energy and power provides heat and electrical and transducer activities and it does have to follow basic energy conservation ideas. But the BJT transistor consumes a small amount of "re-combination" current provided to the base-emitter region in order to activate a larger in-rush of collector current. What's missing here is any discussion of the voltages involved and the rest of the circuit, too, most especially including the energy sources. The BJT transistor draws from those sources and those sources lose energy well in excess of the very tiny local effect by the BJT as one tiny part of a much larger circuit and energy source system.
The conservation law applies to the closed system as a black box. But tiny local increases in energy are possible, so long as those increases come from somewhere else you are ignoring. Just as life itself on Earth may appear to be organization coming from disorganization and violate entropy laws, the fact is that the sun's entropy increases far, far more than any small local decreases in entropy represented by an isolated life form on Earth. You have to take an appropriately complete system when applying conservation laws.