What is the ruling on my extremist secular brother using my computer, knowing that the internet subscription is under his name, and how can I prevent him from browsing extremist websites without causing a strong dispute?

It is not permissible to allow your brother to use the computer to browse secular websites, due to the Almighty's saying: "And do not cooperate in sin and aggression." Enabling him makes you a partner in sin. To rid yourself of this, you must be frank with him about your stance and its creedal motivations, while adhering to gentleness.

If it is not possible to prevent him, then seek another way, and be keen on two matters: understanding the reality of secularism and Islam's ruling on it, and striving to call him to Islam and steadfastness upon it, out of compassion for him. If you cannot call him yourself, then guide someone else to call him. And do not cease supplicating for his guidance.