What is the best way to fix laminate flooring to tiles underneath?
Modern laminate flooring is "floating", meaning you typically have an underlayment (often doubles as a vapor barrier) and then the laminate click-locks together. Finally, you add edging (base board and/or quarter round or toe nail) to lock the edges down.

Screws are NOT how you fix this. You can potentially damage the flooring doing that. Instead, it sounds like you simply have unlevel parts of your floor. What I would do is pull up sections near the unlevel parts and then use a floor leveler concrete to level the underlying tile. Then you put everything back after it's cured up solid. Should remove the "bounce"