Since you didn't provide a picture, or a very helpful description of what you're looking at. I'll try answering your question by explaining how the switch itself works, which will hopefully help you understand the problem better.
Single Pole Single Throw (SPST) Pull Chain Switch
The pull chain switch that controls the light(s), is a single pole single throw (SPST) switch. It has two positions ON (Closed), and OFF (Open). Drawn simply, it would look something like this.
Switch shown in ON (Closed) position.
When the switch is in the ON (Closed) position, current is allowed to flow through the switch, through the light(s), and back to the the source (via neutral).
When the chain is pulled and released, the internal contact rotates 90° (1/4 turn) into the OFF (Open) position.
When the switch is in this position, current is not allowed to flow through the switch, and the light is not lit.
This is why the pull chain switch that controls the light(s) only has two leads.
Single Pole Multiple Throw (SPnT) Pull Chain Switch
The pull chain switch that controls the fan, is a single pole multiple throw switch. It has multiple positions, which allows it to control the speed of the fan. Draw simply, it would look something like this.
Switch shown in OFF (Open) position.
When the chain is pulled and released on this switch, the internal contact rotates 45° (1/8 turn) to the next position.
Another pull, another turn.
Pull again, turn some more.
One final pull brings the switch 180° around, and again to the OFF (Open) position.
By manipulating the output of this switch, the fan is able to whirl around at various speeds depending on the switches position. The number of output leads, will depend on the switch. How those leads are connected to the fan motor, will depend on the fan manufacturer. This simply illustrates the basic principle of how the switch works.
As always electrical work can be dangerous, never be afraid to contact a qualified Electrician
Incandescent lightbulbs won't work very well when conected in series using normal mains voltages. At best they will illuminate dimly.
If your current single 75w lightbulb is on a dedicated 15A circuit, that would be surprising. Perhaps it shares a circuit with one or more wall sockets.
For safe/reliable operation you have to be sure that the total load on that circuit does not exceed 15A. So you need to sum the currents that are likely to be concurrently drawn by each of the parallel loads.
You must also connect your new light-fittings using wire of the construction and gauge specified by local codes (you didnt say if you live in Dhaka, Melbourne or Aberdeen). Connection methods and enclosures (junction boxes) also vary by region.
Using LED lightbulbs will enable you to provide your desired higher light output (lumens) without increasing the power used (watts) or, equivalently, current drawn (amps).
Unless you are happy to spend enough time to learn all this, you might be better off hiring an electrician, it might not be expensive if you do the decorative work, making holes for fittings and cable runs, making good and repainting.
Best Answer
You don't have to run power in a "daisy chain," you can branch from one point to several.
With the sketch you have, switched power coming in at L1, and branching out in two directions, you'll probably have no trouble finding a box large enough to accommodate the wires.