Building strength and good dance technique has to happen long before you get to multiple turns. Once there, increasing your turns is not too difficult. The feeling of spin is controlled through good spotting, musicality, and the same old practise, practise and practise!
One aspect of fast spinning is spotting. You must have a relaxed neck and quick and accurate head movement. Such as, no inclining the head. Inclining happens when you leave the head too long. For more neck flexibility, ice it for 15 minutes after hot showers, and tilt the head sideways for gentle stretches. Do not roll the head around in circles; the neck joints are not designed to do that.
You can work up to more turns by adding half a turn at a time. Do four and a half, which means changing your spot to the back after three and a half turns. That will just kind of bring you around to the back, to catch up to your head. And voila you have done another half a turn. Don't strain, and come down into a soft, controlled stretchy demi plie. Repeat this until it is easy, then add another half, and another half. Little by little as needed, so that you are not compensating for any postural loss, or losing control over your ending position.
I had a teacher once who set pirouette exercises like that, starting with two and a quarter turns. It meant changing your spot to the next wall, and coming around to it. Really easy, not too much difference. You'd do a double, then 2 and a quarter, 2 and a half, 2 and three quarters, then three. Or start with a triple, and do three and a quarter, etc. In pointe shoes, it takes not much more than a thought to add a quarter turn.
Another teacher I had used to say "during your preparation, imagine you are spiralling your spine in the opposite direction to where you are going to turn. Like your inner muscles are twisting to the left, though your shoulders stay square to the front; the prep position doesn't change outwardly. Then when you go up onto releve and turn, you release that twist and it makes you spin. That's a mental trick, and it really works for some people.
Assuming your technique is good, and your postural plumb line is correct, just keep on adding quarter or half turns, and let each addition get easy and natural.