How do you say to "become" in Turkish for eg.

I hope your daughter "becomes" healthy.

Sağol

I´m just a learner, but the verb Olmak means to be, to become, or to happen.

so therefore become is olur, becoming would be oluyor, became is oldu, etc.

And for the last part I´m pretty sure you would just say Geçmiş olsun.