The house is a new build, about 2.5 years old, and all the street wiring is new because the entire street is new. Does this mean the wiring in the house has been done badly?
I'm afraid new build houses are notorious for having very poor telephone extension cabling in them, usually as a result of it being put in by people who don't understand the complexities of RF transmission on twisted pair wiring (which is what xDSL is). Common issues are the use of inappropriate cable (i.e. not twisted pair, but standard alarm cable or similar) or the application of 'star-wiring' (i.e. multiple extensions radiating from a central point).
Good question - how is the extension wire connected to the master socket?
The extension wiring terminals are in the cam-lock connector in the bottom left of your photo and connect to the top (IIRC) row of pins in the test socket. With the faceplate off these are disconnected, and aren't touched by anything else you plug into the test socket since standard BT plugs only have contacts on one side.. The faceplate plug is doubled sided with both sides connected together, so when you plug it in the extension wiring also gets connected to the line.
The reason this harms the broadband, even with your router connected to the master socket, is that the extra wiring can pick up interference and inject it onto the phone line. Additionally, the RF xDSL signal travels up the extension wiring, gets reflected at the unterminated end (or multiple ends if you have star wiring) and then interferes with itself when it gets back to the master socket.
Is this something I can fix? Or do I need to call an engineer?
Yes, you absolutely can fix this (and in fact should not call out an Openreach engineer since problems caused by your own internal wiring attract large charges) subject to basic wiring skills.
The first question is, do you need the extension? If not, simply remove the wires from the cam-lock connector at the bottom left of the photo, replace the faceplate, then connect your router to the master socket and all will be well. If you need the extension, we may be able to explore other possibilities with you.