Now that the question has been updated to include "Thistles" I've revised my below answer a little...
Credit for the "dandelion" part of this answer goes to my mum.
When we (wife and I) moved into our current home 4 years ago, the front and back lawns were covered in dandelions (and other broadleaf weeds). I didn't want to go the herbicide route, and after speaking with my mum, she said the only way to truly get rid of (control) dandelions is to hand remove them (important: you need to remove root n' all) and recommended using a small hand garden trowel for doing so.
I didn't have a "small hand garden trowel", instead I bought a small (and cheapish) builders trowel and used that for 3 years:
Then last year on clearance (paid $1) I picked up a Fiskars Softouch Weeder:
Both of the above hand-tools have worked well in removing dandelions (roots included).
The first three lawn cutting seasons I pulled a lot! of dandelions, at least once a week I would walk the lawns and hand remove all the unwanted plants I saw (all of them went into a plastic bag for offsite disposal).
During last year's lawn cutting season I noticed a big drop-off in the number of unwanted plants I was removing, except for crabgrass (but that's a whole other story).
This years lawn cutting season I've only had to remove a handful of "small" (young) dandelions, this is to be expected as I can't control dandelion seed heads blowing into my garden (though I wish I could).
Yes, getting any unwanted plant under control via an "organic" approach is going to take patience and time, but if you stick with it, you will see (excellent) results.
- And bare in mind, even if you go a "non-organic" route, you're pretty much "forced" to continually use the chosen method if you don't want to see the unwanted plant(s) return.
If you wish to read more about how I approach "organic" lawn care, go here on SE:
Personally I haven't had to deal with thistles in my lawn, that said I do occasionally have to deal with that unwanted plant (weed), along with dandelions and spurge in our street's common ground (flowering) areas that I maintain (I don't mow the grass, a lawn care company takes care of the mowing):
I remove "smaller" thistles using the above tools, basically I treat its removal the same as dandelions ie Remove the unwanted plant, root n' all.
For "large" thistles I use a spade to dig them out (root n' all). After digging them out I back fill the holes with some suitable material and make good the area.