If those are the large thistles like we have here in the east that get up to 6-7' tall and sometimes multiple heads, about all you are going to be able to do is forget the grass for a couple years and mow and spray the field. Our thistles even when cut will bloom about 3" off the ground so mowing is not that effective especially if they already have a head. If you cut those thistle now, they will still seed. I have about 25 acres of field and the thistle got hold when someone brought in bales of hay for cattle and it had thistle in it. I found the most effective method for me because they were all in the same place was to dig them up when they bloom like your picture, put them in the tractor bucket and take them somewhere, pour diesel on them and burn them, but I could do mine in less than a day. The good thing is they are very easy to dig, there's not much root, unless you have a whole field of them which it appears you do. As I said, you may be best to forget the grass for a couple years and work on getting rid of the thistle which may take 2-3 years of mowing 3 times a year and spraying. Mowing will keep them from spreading, but not get rid of them. I've had good luck with some stuff called Crossbow. Your local county agent should be able to help, no one in your community will wan those things spreading. I've seen fields here full of our thistle and they're gone in a few year from spraying. Good luck.
Believe it or not, mowing 3 times a year will not hurt the grass. It will also keep the other weeds from seeding and in 2 years you won't believe how much better your field will look, it will be full of grass instead of weeds.