Using plain screws into just about any material other than wood or metal will not provide a strong support, especially if the weight that the screw is supporting is trying to pull the screw shank straight out (rather than pulling across the shank of the screw).
From your pictures, the surface appears to be plasterboard (or less likely, plaster), which has to be attached to a support structure, either wood, steel or concrete.
Wood
To test if it is wood under the plaster/plasterboard, remove a screw and insert an awl until it just stops.
With some force, try to push the awl in further. If it is wood, the awl should sink in a bit, but not too easily. There should be enough wood to allow you to drill into it and seat the screws fully. You probably need at least 1/2 inch of screw thread in wood for a safe hold on blinds. More is better. Remember that the plasterboard/plaster doesn't count. Try to guage how thick it is (probably either 3/8 or 1/2 inch) and add that to the depth into the wood.
Steel
If the awl doesn't sink in, it may be a metal stud or a metal beam. If you hit something hard and the point can slide easily side to side, it is probably metal. Try to drill a small hole through it with a sharp drill bit. If it is a steel stud, you will be able to with a bit of pressure. You can then use a sheet metal screw to hold the bracket up. If it is a steel beam, you will not really be able to (it is too thick and hard). You will need a whole different mounting system.
Concrete
If the awl stops but can't be slid along a hard metal edge, it may be a concrete support structure. The awl should feel as if it is digging into something a bit gritty. You can mount into concrete but need to use a masonry anchor.
To insert the anchor, you first need to drill a properly sized hole with a special masonry drill bit.
It has a carbide insert in the tip. It is slow going and should be much harder than the plasterboard/plaster. The dust probably will be greyish or tan in color. Be sure not to drill a hole too close to another hole, or the concrete between them may crumble. You then need to insert the anchor and screw the bracket into it.
While it is theoretically possible to get a good enough mount in concrete like this, it is hard. This is especially true when the load is dynamic (moving) the way blinds are.
Plaster/Plasterboard
While it is possible to hang light to moderate loads on plasterboard or plaster, it is really only safe when it is a vertical surface, not an overhead one. You can use various anchors, but not when the force is pulling directly out from the surface rather than down along the surface.
Best Answer
I have 2" blinds from Lowes, probably the same one. Our windows had enough over hang though. If I recall, there are little metal rectangle boxes that get mounted / screwed into the top of the window, and then the the blinds slide into the little metal box to hold it in place. These little metal brackets have numerous holes in them on both the side, and on the top. I think there were four screw holes, one in each corner on the top, and two on the side. If this is indeed the type of bracket your blinds come with, then you would most likely be able to get three screws in place (1 side and 2 in the top). Then repeat that with the other side of the blind.
This should be more than sufficient to hold it in place. I think it would look a little funny though, I would probably go with a 1.5" for looks. But you could use a valance to cover it up and no one would even know.