What's wrong with my code?

Well, you'll want to use more than one equals in your if statement like so

if(correctAnswer === true)

One solution is to put that code in a function, and call the function after each question.

function printResponse(corrrectAnswer) {
  if (correctAnswer = true) {
    finalScore += 1;
    alert ('That is correct. Next Question.');
  } else {
    alert ('Sorry, that is wrong. Next question.');
  }
}

printResponse(true); // correct
printResponse(false); //if they are incorrect

Another solution is to create a function that asks questions, and pass in the question and answer to that function several times

function askQuestion(question, answer) {
  var response = prompt(question);
  printResponse(response.toUpperCase() === answer); //use the function we define earlier to print the response
}

askQuestion("Are rabbits furry?", "YES"); //asks the user if rabbits are furry, and they should answer yes

And if you're really bold, see about creating an array of question and answer pairs, and using a loop to ask all the questions in that array.

your script executes from top to bottom, and the order of execution is the initial alert, the 5 questions, then the conditional statement that increments the score if correct. the conditional statement is only being called once, after the questions. to fix you could expand each question block a little to increment the score if correct (you already check for correctness in each question block), or put the conditional statement in a function and call it after every question, something along those lines.