Not sure what country you're from, but "blockout blinds" are likely what you're looking for. They blockout (essentially) all light and are operated like normal blinds.
If the fabric itself isn't blocking light... You need better material. I have only ever had problem with light leakage in the edges, not in the fabric material.
I believe 'blackout thermal shades' is what to look for.
Installation is key - they need to be oversized, covering the entire window AND the trim, and they need to be carefully installed so that they touch the trim.