You can use match on the string containing the line breaks, and the number of elements in that array should correspond to the number of line breaks.
enteredText = textareaVariableName.val();
numberOfLineBreaks = (enteredText.match(/\n/g)||[]).length;
characterCount = enteredText.length + numberOfLineBreaks;
/\n/g is a regular expression meaning ‘look for the character \n (line break), and do it globally (across the whole string).
The ||[] part is just in case there are no line breaks. Match will return null, so we test the length of an empty array instead to avoid errors.