Electric current is a flow of charge
When the battery is connected up to the bulb to make a complete circuit, there is an electric current everywhere in the circuit. Something flows steadily. That thing is charge, and there can be many different objects that carry the charge.
The current is the same at each point in the single circuit loop – there are no leaks! And no charge accumulates at different points.
The charge originates in the circuit itself. It's already there. That is what it is to be a conductor – to have charged particles that can move when the conductor is connected into a complete circuit.
These charged particles may have other movements as well as drifting steadily, but it is the steady drifting that we'll concentrate on as this movement is the electric current. The charged particles drift steadily in one direction as well as any other movements. The other movements were there before the circuit loop was completed and remain afterwards. The drifting velocity is added to the other velocities.
In metal wires we now know that the charged particles that drift are negative (but it's not at all easy to show this until post-16 study.). That's what's shown in the top pair of diagrams here. But in many other cases, the charged particles that drift are positive (e.g. conduction in nerve cells, electrolysis). We think it's best to be agnostic about the charged particles, but not about the current in the loop: something flows, and the flow is the same at every point in the loop. But we'd suggest representing the direction of conventional charge flow, as in the bottom diagram (where the charge carriers are positive) if you do choose to show charge flows.
The charged particles originate in the circuit itself – when they flow there is a current
In metallic wires the electrons are the moving charged particles and originate in the wires of the circuit. They are simply part of the atoms that make up the battery, wires and bulb. When these components are not connected into a circuit, you might imagine a sea of free electrons buzzing around the fixed array of positive ions (rather like the particles in a gas).
In nerves and electrolysis the current is not carried by electrons. We'll refer to electric currents in terms of a flow of charge, as this covers all cases.