How can I get better at cycling fast when wearing a heavy bag?

In the physical world of a cycling, you are the motor. Like any vehicle, to carry more weight, faster, you need a more powerful motor. That means greater torque and more horsepower.
In engineering terms, you would try to create a bigger bang in the cylinder and increase the rev range. In cycling terms, that means more strength in the pedal push and increase the cadence (pedal rpm). Most normal humans find pedaling at more than 100 rpm very difficult and even pro-cyclists struggle above 120 rpm. So it's best to focus on the pedal power, and then gear up, to suit your own maximum cadence. (I'll go into increasing cadence in a bit.)
Here are some tips about getting at cycling fast when wearing a heavy bag.


Something about the pedal
Most of the power used to turn a pedal comes from the action of effectively standing on the pedal, so simple squats will help immensely here. However, the biggest gain will come from having the stamina to perform this action repeatedly, for long periods of time. I like to think of this process as running. Yes, it's as simple as that. Become a better runner and you WILL become a better cyclist. Where possible, run up steep hills or steps. This will really build the power and stamina.
Back to cadence
It's important to note that the perfect cadence is different for everybody. And only a man in a white coat, with lots of equipment is going to tell you yours. However, getting your cadence up to 90–100 rpm is going to work for most people. The trick with building cadence is to do it gradually and a long period of time. Using an exercise bike is much easier, because you can see exactly what you're doing but it's also easy to get carried away, as you don't really experience hills and headwinds.
Gets the right gearing on the bike
It is the easiest. It's about getting the right gearing on your bike. The extra weight is going to make you faster downhill, so you need a long gear for that. As you get stronger you will become less reliant on the shorter gears but while you're building, you may need to consider a more MTB focused setup. I.e. Small front and very large big ring on the back. Your local bike shop should be happy to help with this.


Find an aerodynamic position.
The best way to improve your speed is to find an aerodynamic position that is still physiologically efficient. You can't ride fast if your hands are numb or your knees are banging, or your chest is constricted. Again, moving the cargo off your body is a great solution.
Don’t wear a heavy bag
The only time it's remotely practical to wear your luggage while cycling is off road since the trips tend to be shorter and you don't want the panniers to clip obstacles like trees and rocks. Other than that, you should figure out a means of fastening your luggage to the bicycle, not your back. Wearing your luggage makes you sweat and puts unnecessary strain on muscles not involved in cycling. You say, "Heavy bag" so that implies panniers: install a rear rack on the bike and buy a pair of large panniers. If that's not enough, you do the same thing in the front though here you need to keep the weight to a minimum so as not to destroy the handling. There are other places to clip luggage: in front of the handlebars, on top of the rear rack, inside the rear triangle and of course behind the saddle.
Give up unnecessary luggage
Wearing an unnecessary luggage is always going to be a drag. For occasional, short rides it's tolerable. But if you're looking to ride frequent long distances, you're going to need to transfer your cargo somewhere else on your bike. This will improve your ride in a number of ways: your body will not tire as quickly from supporting the burden (lower back and arms), your bike will handle better because the center of gravity is lower, and you may ride marginally faster because you reduce your aerodynamic drag. If you transfer your cargo to a bike trailer, you may also have fewer mechanical failures, because the load is spread out over more wheels.


In a saddle bag
There are literally hundreds of small saddlebags on the market, intended to provide a safe home for your tools, spares, wallet and keys. Bigger versions will take spare clothing too if the weather’s changeable or you’re out for an adventure. Lightweight cycle tourists have always used bigger saddlebags; the latest designs to come out of the bike packing and adventure racing world are big enough to swallow half your gear for an overnight. It is tidy and could contents protected from the elements. What’s more, it always there ready for a ride.
Bike Trailers
There’s a great deal of price range in trailers too. Few of them are cheap, however, but good deals can be found on used trailers through Craigslist. Cloth trailers can carry LOTS of weight, more groceries than you probably ever buy in one trip, but they are generally not sturdy enough for furniture. And if they are not meant for children, don’t put your children in them! Trailers that are specifically for children are a little more expensive, and often have seats, belts and orange safety flags. Some trailers include a special hitch attached to your seat post, others can be attached to any bike very quickly, with no tools. For very, very heavy things, there are cargo trailers. These have solid bottoms and are often rated up to hundreds of pounds.
Panniers
The quintessential on-bike bags, panniers sit either side of your bike so a large load is balanced. However, there’s nothing to stop you using just one if all you need is to carry a day’s worth of office stuff. You’ll need a rack to hang them on, at which point you have to decide whether you want that rack on the back of the bike or on the fork. A rear rack is traditional if it’s going to be the only place you hang luggage, but there’s a school of thought that says low-rider front panniers are actually better. Very heavy rear panniers can dramatically affect the handling of your bike, causing the back end to wag around. Because the weight is close to the steering axis of the bike, low-rider front panniers have far less effect. You only get that advantage with a low-rider rack though; large, heavily-laden front panniers up high are a bad idea. And it is comfortable and has large capacity.