Joe Jonas and Sophie Turner Appear To Have Child Custody Arrangement

Joe Jonas and Sophie Turner appear to be making progress in their bitter child custody dispute ... because it looks like they've come to an agreement on a time-share with the kids.

The Jonas Brothers member spent Monday morning with his two daughters at a park in New York City before dropping them off with his estranged wife.

As you can see, Joe was all smiles as he left Taylor Swift's place ... that's where Sophie is staying while she's in the Big Apple.

The handoff makes it seem like Joe and Sophie hammered out an agreement on child custody ... as we first reported, the former couple is deep in mediation to settle their nasty divorce.

Joe and Sophie have been spending a lot of time this month trying to reach a divorce settlement ... remember, they have a prenup and one of the provisions is that they go to mediation to settle all outstanding issues related to their divorce.

As we reported ... we've been told they are both cool with joint custody, provided they can agree on a structure, and this makes it look like they've done just that.