There is only a very small error in your code, from what I can see. The last if statement, which includes your “Game Over” string, is indented incorrectly. Move it left one block:
  else:
    print "You missed my battleship!"
    board[guess_row][guess_col] = "X"
if turn == 3:
  print "Game Over"
  # Print (turn + 1) here!
  print_board(board)
Here is my complete code from that exercise:
from random import randint
board = []
for x in range(5):
    board.append(["O"] * 5)
def print_board(board):
    for row in board:
        print " ".join(row)
print "Let's play Battleship!"
print_board(board)
def random_row(board):
    return randint(0, len(board) - 1)
def random_col(board):
    return randint(0, len(board[0]) - 1)
ship_row = random_row(board)
ship_col = random_col(board)
print ship_row
print ship_col
for turn in range(4):
    print "Turn", turn + 1
    guess_row = int(raw_input("Guess Row:"))
    guess_col = int(raw_input("Guess Col:"))
    if guess_row == ship_row and guess_col == ship_col:
        print "Congratulations! You sunk my battleship!"
    else:
        if (guess_row < 0 or guess_row > 4) or (guess_col < 0 or guess_col > 4):
            print "Oops, that's not even in the ocean."
        elif(board[guess_row][guess_col] == "X"):
            print "You guessed that one already."
        else:
            print "You missed my battleship!"
            board[guess_row][guess_col] = "X"
        print_board(board)
    if turn == 3:
        print "Game Over"
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solved Codecademy Battleship! Python