在…下【zài … xià】

在…下 names the conditions, help or pressure something happens under. The noun inside is abstract, usually with 的 before it, and the whole phrase goes before the verb. It is written and formal.

The nouns that fill it are a small set: 帮助, 支持, 鼓励, 指导, 影响, 压力, 条件, 情况, 要求, 领导.

With 情况 and 条件

在这种情况下 and 在…条件下 set the scene for a rule or a decision: "in this situation", "under these conditions". They are common in instructions and reports.

The phrase comes first

English puts "with my teacher's help" at either end. The Chinese phrase goes at the start of the sentence or after the subject, never after the verb. And 下 can't be left off.

• × 我通过了考试在老师的帮助下 → 在老师的帮助下，我通过了考试
• × 在老师的帮助，我通过了考试 → 在老师的帮助下，我通过了考试

在桌子下 with a physical object is a place, "under the table". 在…上 and 在…中 are the frames for "in the area of".