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 charges originate in the circuit itself. They are already there. That is what it is to be an electrical conductor — to have charges that can move when the conductor is connected into a complete circuit.
These charges may have other movements as well as drifting steadily, but it is the steady drifting that we'll concentrate on because this movement is the electric current. The charges 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 is added to the other movements.
In metal wires we now know that the charges 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 charges that drift are positive (e.g. conduction in nerve cells, electrolysis). We think it's best to be agnostic about the charges, 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 think about charge flows.
The current is the same at each point in the single circuit loop — there are no leaks! And no charge accumulates at different points.
In metallic wires the electrons are the moving charges 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 their insides as a gas of free electrons pinging around the fixed grid of positive ions.
In nerves and electrolysis the current is not carried by electrons. To have a general model to reason with, we suggest you think of electric currents in terms of a flow of charges, because this covers all cases.