The courts will start from the presumption that the child should have a relationship with both parents. If the evidence is sufficient, it is possible to challenge that presumption, but it does take good evidence and quite a lot of time.
My exH wasn't very violent physically, but I would categorise him as emotionally abusive. Custard is right in saying that this is an elastic term - in my case, exH quite genuinely tried to persuade 5-year old dd that I (her main carer) wanted to kill her, which, as you can imagine, was pretty destabilising for her. He also made numerous false allegations to SS and GPs and A&E that I was injuring her - there are several medical reports saying "X alleged injury to child, but none was found". it was tough on DD, who had lots of interviews with social workers and felt she was betraying a parent whatever she said. Dd's school was pretty concerned about her emotional wellbeing.
With all this evidence, I was eventually successful in court and got the contact reduced. It's still unsupervised, but it's shorter, and the court order allows me to reduce it further if exH is acting up.
Despite everything, the unsupervised contact is sort of okay now, several years down the line. Dd is older and can understand that her father says things based on anger that are not necessarily based on fact. I wouldn't say he's a great father, but there is a relationship with some positive aspects. Of course children need to be protected from abuse, but it can be worth salvaging some bit of that parental relationship if it's reasonably possible. Emotions do cool down a bit over time.
There's rarely a perfect outcome when it comes contact in these cases. Courts try to bodge together the least worst outcome for the child.