In a circuit yes. The charge carriers in the wires are electrons.
Let us imagine a series circuit consisting of:
1. A voltage source.
2. In series with a resistor.
## I_{series} = \dfrac{ V_{source} }{ R_{1} } ##
in the standard model of any electric circuit with some components, an increased voltage increases the current in the circuit. Current is defined as being the rate of flow of charge carriers.
If more charge carriers are flowing past an imaginary plane in a section of wire before the terminal of the ressitor, then indeed the speed of charge carriers, has increased.
In fact, there is a relationship between current and drift velocity.
https://en.wikipedia.org/wiki/Drift_velocity
So, you are right, under certain conditions of course.
Again, current is defined as the flow of charged particles at a point or plane. If current increases, then the flow of charge carriers has increased.
So, technically, the flow of charge in a series circuit is the same, ie . the speed of the charge carriers (electrons) is the same in a series circuit. But in a parallel circuit, the flow of these charge carriers (electrons) differs, as current in branches is different.
Resistance is also defined as opposition to flow of charge. But charges move very slowly, about a few CM or MM (i don't remember) in a circuit. It is the energy transfer that is faster.
But of course, in an alternating current, charges fluctuate or jump around back and forth.