Choose the correct option after joining the given sentences.I wanted…

The correct answer is Option (2) → I wanted to go for a walk but it started raining.

Joining two clauses with a conjunction of contrast:

• But is a coordinating conjunction that joins two clauses whose ideas are opposed: the second goes against what the first leads us to expect.
• This sentence: the speaker wanted to go for a walk, and the rain came in the way of that wish, so "but" joins the two.
• Complete sentence: I wanted to go for a walk but it started raining.
  Example: She studied hard, but she missed the first rank.

Why the Other Options are Incorrect?

Option 1: I wanted to go for a walk while it started raining. — Incorrect. While links two actions going on at the same time; the contrast between the wish and the rain is lost.
Example: He read a book while he waited for the train.

Option 3: I wanted to go for a walk so it started raining. — Incorrect. So introduces a result; it would make the wish to walk the cause of the rain.
Example: It was late, so we took a taxi.

Option 4: I wanted to go for a walk lest it started raining. — Incorrect.

• Lest means "for fear that" and states a precaution, not a contrast.
• It is followed by "should" or the base form of the verb (lest it rain), not by the past tense "started".
  Example: Walk carefully lest you slip.

Contrast: but shows contrast, so shows result, while shows time, and lest shows fear.