Learn Turkish: Dar pantolonlar artık demode; tersine, salaş kıyafetler giyiyoruz. - Tight pants are out of fashion now; on the contrary, we wear baggy clothes.

Yes, both tersine and aksine mean 'on the contrary' and are completely interchangeable in this sentence. They are used to contrast the first statement (tight pants being out of fashion) with a directly opposite action (wearing loose clothes). Notice how the conjunction is placed at the beginning of the second clause to pivot the meaning.

What exactly does salaş mean in this context? Doesn't it usually mean 'shabby'?

While salaş can mean 'shabby' or 'ramshackle' when talking about places or buildings, in the context of clothing it means 'baggy,' 'loose-fitting,' or 'casual.' It acts as the direct opposite of dar (tight) in this sentence.

Why is there no 'to be' suffix after demode?

In Turkish, third-person statements often drop the factual 'to be' suffix (-dir), especially in everyday language. You are completely grammatically correct to say demodedir here, but leaving it as just demode is very common and sounds completely natural.

Why is giyiyoruz (present continuous) used instead of the simple present tense if this is a general habit?