“if your high school transcript includes suspensions and the exact words bullying and sexual harassment, you’ll have to explain it.”
The wording on the transcript, or whether the discipline is on your transcript at all, is irrelevant. The Common App and almost all highly competitive universities will ask if you have ever been suspended. You will need to say yes when asked, even if your high school has a policy of not disclosing these events. That said, @melvin123 is right that the severity of the offense and the way you explain it play a big role in determining the impact of the suspension on your applications.