Thank you so much! I the circuit started working after I switched the resistors.
Honestly, thank you so so much. My teachers at school had no idea what they were looking at and it was incredibly frustrating.
For future reference, how did you identify what was wrong? I'm a beginner so any and all advice would be appreciated.
Oh dear - the teachers do not know everything!
I do love the irony of that website name - "rookie electronics". We often refer here to the "instructables" website where many projects are well-meaning but actually simply wrong and sometimes quite dangerously so.
OK. The transistor's purpose is to amplify a small current to control a larger one. A smaller current generally corresponds to a larger resistance (Ohm's law, but let's not get int an argument about that ), so you expect the base resistor to be larger than the collector resistor by a minimum of ten times. The gain of the transistor is going to be at least ten (but there are further tricks there which we need not be concerned about in this application).
The problem with the swapped resistors is twofold. One is that the 47k resistor may not be sufficient in itself, to effectively "pull up" the voltage to the servo. I don't know exactly what the input impedance of the servo is (someone else may have an opinion) but I'm betting it is actually less than 47k.
Secondly, bipolar transistors suffer from "charge storage" so that if you feed a lot of current into the base, far more than necessary to perform the amplification function, this charge is stored in the base-emitter junction and it takes some time for that charge to dissipate during which time the transistor continues to conduct even thought there is no voltage on the base. This time is measured in hundreds of microseconds, but that happens to be the sort of time which is critical in control of a servo. And it is made worse by the too-high collector resistor.
So there you have it. I learned this - the hard way - some forty-odd years ago, and transistors have not changed in the meantime though we now use FETs for preference, and they have some limitations with charge storage for somewhat similar reasons.
Protocol here dictates that comments/ replies - even compliments - be on the forum itself rather than private not merely because it makes me feel better ( ) but so the nature of the problem and its solution is confirmed. You can go back to the first post and modify its title to include "(Solved)".