This is why it's generally a better idea to book directly with the airline.

When you book via an agent or third party, your contract is with them, so you are subject to their fees and charges. They can (and do) have more restrictive conditions and much higher fees than other agents, or the airline in question. Presumably you booked with the agent, rather than directly with the airline, because the agent was offering a cheaper price. The price to pay for a cheaper ticket, however, is usually more restrictive terms and fees which bite you when you seek to make a voluntary change. This is how these agents are able to undercut others' prices - by fleecing some customers! Note also that another "trick" is for agencies to charge far more for seat selection and baggage than the airline themselves would, if these ancillaries were booked directly with the airline in question. They may also charge fees for other spurious services, such as checking you in automatically, or sending you SMS confirmations.

You should normally expect to pay 4 change fees. You would only normally expect to pay 16 change fees if you changed each sector on each ticket separately and individually - i.e. you rang up 16 separate times to change a single flight sector on a single ticket on each separate phonecall.
Instead, you should change all 4 sectors on all 4 tickets in a single transaction, leading to 4 change fees/

On top of this, you will have to pay any increase in fare (but won't get any refund or credit if the new flights actually cost less than the original fare) and any service or handling charge that the agency may apply for doing this (which may be levied in the form of long call times on a premium-rate phone number)