Disclaimer: This article has some broad generalizations that will have exceptions to what is stated. This is the only way I can even attempt to address the topic. I've looked for scholarly articles published on peer reviewed journals for years. I don't know of any. Years ago, a JW and an apologist for the WTS named Carr wrote a slanted article on mental health in favor of JW's.
Since JW's are a closed group, there is no way of accurately knowing the degree of mental illness among its members. However, we can look at the symptoms among its members to come up with some educated guesses. I will draw on my experience and admit in advance this is a subjective viewpoint from an experiencer.
In my last congregation I attended, I witnesses forms of depression from families living in poverty with no prospects of a better future. Continuing education or specialized training for a technical job was forbidden if it interfered with making meetings or field service. This double bind (damn if you do damn if you don't) filled the lives of many members. When husbands (many elders) were emotionally or mentally abusive with their family members, there was no relief. Wives were advised to be submissive pray and leave it in God's hands.
Since JW's are such a sexually repressive group, issues of infidelity and prohibited sexual acts were rampant. When elders or ministerial servants were the predators, the victims were told to leave it in God's hands.
Some portion of the demographic were members with existing mental illness pre-conversion. These people clung to the faith as an antidote for their illness fantasizing an idyllic future in the paradise. In my opinion these people's illness was exacerbated from the binds the religion created for them. In our congregation, I head of one lady who was disfellowshipped for sex out of marriage and in her meeting with the elders, she broke down and went into a fetal position crying uncontrollably. She was having an emotional psychotic breakdown.
One lady I knew in another congregation suffered from bipolar depression. In one of her manic episodes, she sold her house and went to serve where the need was greater. This could have been avoided if she had been encouraged to seek medical and psychological help. When she cycled back to normal, she was broke with no one to turn to for help.
In my experience, a very large percentage of personal problems among JW members had a psychological explanation that psychotherapy could have solved. This option was strictly forbidden and anyone who turned to professional help was viewed with suspicion.
Do JW's cause mental illness? Yes, from the hypnotic alternate reality they construct with hypnotic language. When reality clashes with this unrealistic magical thinking mindset, the witness is told he has double down on having more faith and being submissive to the elders.