Does forgiveness in Islam mean turning the other cheek?
No. Islam allows a wrong to be answered in equal measure, but never beyond it. Yet it recommends forgiving and returning evil with good whenever that brings the other person to reform. This is a noble choice, not a compulsion.
The image of the other cheek comes from a teaching of Jesus (as), which many Christians understand as a fixed duty. Islam takes a different path. It does not say that you should let yourself be struck a second time. It says that you may answer a wrong in equal measure. If someone shouts at you, you may raise your voice too. But whoever gives back more than he received becomes a wrongdoer himself.
Alongside this right stands a recommendation. Whoever forgives and thereby moves the other person to change for the better has a sure reward with God. So the key is not weakness but wisdom. You ask yourself what will truly change the other person. Sometimes it is firmness, but often it is kindness. The verse above promises that an enemy can become a warm friend if you repel evil with what is best. Do good to the one who harms you, in order to win him over.
The Holy Qur’an also gives a clear attitude towards insults. If someone abuses you, you do not have to stay and take more of it. You may walk away with a greeting of peace. Later, when you are alone and still boiling inside, you turn that anger into prayer. You ask God to guide the other person to the right path. This is not a show of submission in front of him, because it happens in private. It purifies your own heart and can change the other person as well.
This teaching is strong advice, not a burden that has to break you. If you cannot manage to forgive and only answer in equal measure, you have done nothing wrong. But if you can, you are choosing the higher path. As the Holy Prophet Muhammad (saw) taught, true strength shows itself precisely in keeping control of yourself when you are angry.