Content warning: This story discusses child pornography.
His sister Jill and her husband, Derick Dillard, shared a statement on their website following the verdict.
"Today was difficult for our family," they began. "Our hearts go out to the victims of child abuse or any kind of exploitation."
Jill and Derick said they're "thankful for the hard work" of law enforcement, prosecutors and "all others involved who save kids and hold accountable those responsible for their abuse," adding, "Nobody is above the law."
"Josh's actions have rippled far beyond the epicenter of the offense itself," continued the Dillards, who have distanced themselves from the Duggars in recent years. "Children have scars, but his family is also suffering the fallout of his actions."
They said their "hearts are sensitive to the pains" of Josh's wife, Anna, and their seven children.
"This trial has felt more like a funeral than anything else," the couple wrote. "Josh's family has a long road ahead. We stand with them, we are praying for them, and we will seek to support them however we can during this dark time."
Josh's cousin Amy also spoke out on Twitter Thursday, Dec. 9, writing, "May the juror's and Judge Brooks seek healing / counseling from all that was exposed to them. May the children grow up with real Godly men as father figures who will guide and protect them. May the people who were once in disbelief have their eyes opened for the first time."
"May the daughters who were abused feel validated. You are truly beautiful and worthy of love. May there be intense counseling / healing. May there be wisdom for all those involved on how to move forward," she added in a separate tweet. "Justice has been served."
Amy's mother, Deanna Duggar, is one of Jim Bob Duggar's sisters, and raised the 35-year-old outside of the fundamentalist church.
Josh is currently in custody and awaiting sentencing. He faces up to 40 years in prison total, as well as fines up to $500,000.
In a statement issued after the conviction, Josh's attorneys Justin Gelfand, Ian Murphy and Travis Story told E! News, "We appreciate the jury's lengthy deliberations. We respect the jury's verdict, and we intend to appeal."
Eight months ago, Josh was arrested on suspicion of using the internet to download child sexual abuse material. He pleaded not guilty to the two charges and was later released on bail.
At the time, Jim Bob and Michelle Duggar said in part that they hope the "truth, no matter what it is, will come to light."
In a new statement, issued exclusively to E! News on Thursday, Josh's parents said that their "hearts and prayers are with anyone who has ever been harmed through CSAM [Child Sexual Abuse Material]."
"In the days ahead, we will do all we can to surround our daughter-in-law Anna and their children with love and support," Jim Bob and Michelle said. "As parents, we will never stop praying for Joshua, and loving him, as we do all of our children. In each of life's circumstances, we place our trust in God. He is our source of strength and refuge. Thank you for your prayers."