What were those diet pills?

I don't have an in-universe answer, but in the real world methamphetamines and related drugs (uppers, speed, meth, etc...) cause suppression in appetite and increase of physical activity usually leading to weight loss. Prescription medications for ADHD are of that class and have those side effects. I believe (haven't researched) diet pills like phen phen or whatever they were called, are chemically similar and cause similar effects. Those types of unregulated drugs (in the US) were controversial because of health side effects and I believe were pulled from market.

The real-life Rx medication is called Adipex, which is molecularly similar to Adderall (an amphetamine used to treat ADD/ADHD). =)

Amphetamine, speed, Adderall, etc.

It was Adderall used for treating ADHD but is known to make you lose weight as a major side effect is lack of appetite.

Although Adderall is barely prescribed anymore nowadays as it has been replaced by Vyvanse, another amphetamine-based drug which is far more powerful but works better and has less negative effects, but that too makes you lose weight.

Any amphetamine will make you lose weight.

The are Adderall which is used to treat narcolepsy and ADHD.

From what I gather, they were most likely Desoxyns- prescription methamphetamine. My mother was prescribed Desoxyn when she was 14 years-old (1971) for weight loss. She described having the same side effects as Sarah (before she started to abuse them). But most amphetamines- Desoxyn, Dexedrine, Adderall, etcetera- create that euphoric, on top of the world feeling and suppress appetite.

What were those diet pills?

As others have said, some kind of amphetamine-like stimulant.

The other thing is she was also prescribed a sedative to help her sleep.

As she was taking the stimulant, the effect wears off after a while, so she was taking more than what was prescribed. That was making her really wired and she was taking the sedative at the same time to calm herself down and ended up taking more and more of both of them.

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