What is the ruling on praying behind an Imam who makes many mistakes and forgets often, yet insists on leading the prayer, and is one sinful for leaving prayer behind him and going to another mosque?

The Sunnah is that the most knowledgeable of the people in the Book of Allah should lead them in prayer. If they are equal in that, then the most knowledgeable of them in the Sunnah. Then, the one who migrated earliest, then the one who embraced Islam earliest. Merely wearing a jilbab and a turban does not make a person more deserving of leading the prayer.

The prayer is valid behind an Imam if his mistake is not in Al-Fatiha, or if it is in Al-Fatiha but does not change the meaning. In such a case, it is legislated to correct the mistake, and it becomes obligatory if it is in Al-Fatiha.

You are not obligated to pray in the aforementioned mosque. It is advisable to advise the Imam to step aside if there is someone present who is better than him in recitation.

If the mistake is in Al-Fatiha and it changes the meaning, then prayer behind him is not valid if he intentionally made the mistake or if he was able to learn but did not. The mosque community should be alerted to this.