What tutorial you using?
Check the process of storing the highscore. I suspect you will find the problem there.
Also run it in debug mode and see what is being stored or isn't being stored
i use Construct 2 video tutorial,and by the way,that's all same with the tutorial.
and now i have tested it again,i want ask something,Web Storage save all of highscore right?
and the local key values must different with first time local key?
ex :
First time : HighScore
Second time : HighScore2 => is this problem?