There are those who argue that for churches to grow it is necessary for pastors to give up pastoral care. I reject that. I think that being involved in some pastoral care is by definition part of being a pastor.
Having said that, they are correct that pastors cannot provide all of the care. Not only are there other responsibilities for the pastor, pastors are meant to equip the saints for ministry. Training laypeople to care for others in the congregation is part of pastoral care.
In our church, we have what we call Care Teams. We have divided the congregation into groups under a Care Team Leader. The Care Team Leader keeps tabs on the people on their list, not providing counselling, but being a support and letting me know when I need to step in.
It can be a lot of work to get such a structure in place but it will pay off. It will be a blessing for the people both providing and receiving the care.