First of all, in turn.h you should declare the enum using a typedef like this:

#ifndef TURN_H
#define TURN_H

typedef enum {
    WHITE, 
    BLACK
} Turns;

typedef struct {
    Turns turn;
} Turn;

void setTurn(Turn *t, Turns newTurn);
Turns getTurn(Turn *t);
void changeTurn(Turn *t);
Turn turn;       // This allows main to see the struct declared in turn.cpp

#endif

As you can see in code above, to solve your problem, I also added Turn turn; to turn.h. This allows main.cpp to see the struct you want to declare in turn.cpp. Then, remove extern Turn turn; from turn.cpp because it has no effect.

Then, modify turn.cpp like this:

#include "turn.h"

Turn turn;

void setTurn(Turn *t, enum Turns newTurn) {
    t->turn = newTurn;
}

enum Turns getTurn(Turn *t) {
    return t->turn;
}

void changeTurn(Turn *t) {
    if (t->turn == WHITE) {
        t->turn = BLACK;
    } else {
        t->turn = WHITE;
    }
}

Note:
The keyword extern looks for declarations of the variable or function passed as its argument outside the file in which the extern is. Putting it inside turn.cpp isn't useful because Turn turn; is declared in itself. You should put the extern if the file which doesn't have the declaration inside,

I want to specify that typedef is not mandatory for the enum:

#ifndef TURN_H
#define TURN_H

enum Turns {
    WHITE, 
    BLACK
};

typedef struct {
    enum Turns turn;
} Turn;

void setTurn(Turn *t, enum Turns newTurn);
enum Turns getTurn(Turn *t);
void changeTurn(Turn *t);
Turn turn;

#endif

As you can see, it works also without typedef.
However, everytime you declare a Turns you have to use the syntax enum Turns, which is verbose and less readable. If you want to modify all declarations of the enum across all your files.