Is it normal that my periods are so short and random?

It is not normal actually
But don't worry
if you started to take care of the problem you have nothing to worry about .. Usually it is something manageable with treatments
You have to see a doctor

As for your menarche starting at 15 .. It is normal if your breast started to develop no more than 2 years before menarche starts

Females start having their periods at 9 or 10 years old in this society.
Starting at 15 years old is unheard of nowadays.
That's why doctors recommend that girls age 9 get their 3 doses of Gardasil vaccines , before they start having sex , so the HPV virus vaccines take effect before they start having sex.
Females started having their periods at 14 and 15 years old in the early 1970's.
Maybe because you are so heavy for your age might have something to do with your irregular and light flowing.
You should be on a timely monthly schedule for 5 days a month with a moderate to heavy flow from the 2 nd - 4 th day , non fail at 15 years old.
It shouldn't be irregular or light flowing until you are getting close to menopause when you are about 47 - 50 years old.
You should be in full menopause mode at 52 or 53 years old.
Menopause can last up to 7 years.
If you stop going through menopause yn that 7 year stretch and start up again , the 7 year time frame starts all over again from day one , it doesnt pick up where you left off when menopause stopped.
Go to a woman { Cunt , Twat , Suzy } finger doctor and get it checked out.
Are you sure that you are not pregnant ?

Go see a doctor just to put your mind at ease. If there is a problem it is better to find out now than leave it and find out you have a bigger problem later.

It may be worth going to see someone. It could well be nothing, but I would imagine that after two years, your period should be established and regular unless there's something up.