An electrician is telling me I have to move the bare wires and it doesn't look like that's the case in the code.
The electrician is correct about that. You have shown us a picture of a sub-panel with a four-wire (hot/hot/neutral/ground) feed. There is no main breaker in this panel. There is a main breaker somewhere else, and neutral and ground are bonded together there. In this panel, neutral wires and ground wires must be kept strictly separate - neutral wires to either of the two bars just above the breakers fed by the insulated neutral wire, grounds to the ground bar on the far left fed by the bare ground wire.
You have made a very dangerous error: if the neutral between the main breaker and this panel somehow gets disconnected, all the grounds that you've connected to the neutral bar will be hot and someone could be killed by touching a grounded metal appliance casing (and no downstream GFCI could save them because GFCIs only monitor and interrupt hots and neutral - not ground). In the case of that kind of failure, everything that should be perfectly safe (grounded) becomes deadly.
I'm very suspect of this electrician because he was telling me I need a 50 amp breaker for a 50 amp EV charger and I was almost positive I needed a 60 amp breaker and I needed to set the charger to 48 amps max.
You are correct on the basic principle. But we need more details (EVSE model, wire type and size, whether you have hard-wired or used a receptacle, etc.) to know if that installation as a whole is correct. You also need to perform a load calculation to know if you actually have the service capacity to add a new 48A continuous load.