Is it normal for intercourse to start your periods?

No, sexual intercourse cannot start your periods.

When you menstruate is determined by your menstrual cycle, a domino effect of hormonal changes effecting your reproductive organs. If your period is due it can induce your period, but that's it.

Your cycle is not changed by sex, nor can sex allow you to speed-up your cycle or skip phases of your cycle - if your cycle was impacted by sex then how would anyone ever get pregnant?!