Hello D&L,
It would be easier to explain this if we had more context. Some of the words, like circumstances and equally, are common words not used idiomatically, so I don't know why they are causing you trouble.
The courts - custody is decided in the court, by the judge or other official.
Custody - who is determined to be repsonsible for a child (or the children) in a divorce. We use the phrase "custody is awarded to..." or "they were awarded joint custody"
Circumtances - You can use your dictionary for this. I don't know what the context is, so I don't know what the circumstances are.
"The tide is turning" - an idom to say things are changing
equally able - just as good at something as someone else - in this case, men are just as good at being parents as women
(I'm surpised when custody goes to any single parent these days - most courts award joint custody unless there is something seriously wrong with one parent.)
EDIT: Sorry to have been so slow. I see Clive has beaten me to the punch. However, since I'm divorced from my children's father, if you have specific questions about how this works, please let me know.