Learn Dutch: Wij denken dat het niet mogelijk is. - We think that it is not possible.

This is because of the word dat (that). In Dutch, dat introduces a subclause (a dependent part of the sentence). Whenever you use dat to connect thoughts, it acts like a magnet and pulls the working verb in that second part (here, is) all the way to the end of the sentence.

Can I leave out the word dat like we often drop 'that' in English?

No, you cannot. In English, it is perfectly fine to say 'We think it is not possible' and drop the word 'that'. But in Dutch, you must always include the linking word dat when connecting these kinds of clauses.

Why does niet come before mogelijk instead of at the end of the sentence?

When you use niet (not) to negate a specific adjective (like mogelijk), it goes directly before that adjective. It translates directly to 'not possible' (niet mogelijk).

Could I say we denken instead of wij denken here?

Yes, absolutely! Both we and wij mean 'we'. Wij is the emphasized form. If you want to stress that we think it is impossible (implying someone else might think it is possible), use wij. For a neutral, everyday statement, we is usually preferred.