Watch flamenco in a pena
Unfortunately, almost all Flamenco shows in Spain, especially in larger towns, will have many tourists. But if you are willing to be patient and keep a watch out for last minute, late at night performances, then you should go to a flamenco pena - almost every neighborhood has one - the local club where residents of the area go for their entertainment.
Pena performances are by amateurs
You may need to pay a small entry charge to enter; there is also typically some tapas and drinks for sale. Be sure to carry cash. Of course, some penas host free flamenco shows.
Here is a useful list of flamenco penas in Seville. My suggestion is to go as away from the center of Seville as you can, for the most 'real' experience of Flamenco.
Visit Seville during festivals
During flamenco festivals, you can catch authentic flamenco performances.
Tip - Many would suggest going to La Carboneria in Santa Cruz but the funny thing is that with so many suggestions you can see on the internet all pointing to Carboneria, it is now full of tourists.