The difference between In general and Normally

When used as adverbs, in general means in the general case, whereas normally means under normal conditions or circumstances.

check bellow for the other definitions of In general and Normally

In general as an adverb:

In the general case; without further assumption; without qualification; in all respects.
In general as an adverb:

As a rule; usually.

Normally as an adverb:

Under normal conditions or circumstances; usually; most of the time

Examples:

"Normally, I eat breakfast at 6am, but today, I got up late and didn't eat until 9."
Normally as an adverb:

In the expected or customary manner.

Examples:

"Lisa ate normally, until she realised that she was late for [[choir]], when she sped up."
Normally as an adverb:

To a usual or customary extent or degree.

Examples:

"He was abnormally agitated, she only normally so."
Normally as an adverb (mathematics, statistics):

In the manner of a variable with a Gaussian distribution.