My iphone 5c's battery has inflated? The screen is almost coming off?

Take it into Apple for them to examine it. It appears the battery is expanding.

Is it still in warranty? If so Apple will replace it. Also if you live close to an Apple Store it may be worth it to take it in any way, I have seen them "help" a lot of users anyway. However you said that you already have a new one so disposing it can may be the best option.

I just wanted to add a friendly reminder that you can't throw electronics in the regular trash. -AJ

Thanks for the picture ... speaks loudly.

http://www.fire.tc.faa.gov/pdf/systems/Lithium-ion_battery_04112006.pdf

As for fire from lithium batteries .. it is said to have happened. I hear it illegal to transport in passenger airplanes now as people die from such a confluence of events ... altitude high, pressure low, radiation high, batteries package together making for a critical mass ... other factors in discovery ... they gather the technical evidence from the crash test dummies as the cases develop from the consumer market.

😢

You must be one of the most misinformed people in the world. Or you make this stuff up. It is not illegal. If it were illegal you could not take a computer, tablet, electronic game or any cell phone on any airplane. And what about the flight crew, who all now have iPads? Is it illegal for them to have those iPads that have their flight plans and charts on them? And remember that all modern airplanes have Lithium batteries as backup power. NO ONE has died from any confluence of events. Altitude, pressure and radiation are also irrelevant, and you apparently don't know what "critical mass" actually means. The only point where you are partially correct is that airlines now do not permit Lithium batteries to be transported as cargo on passenger airplanes, out of an "abundance of caution," because ONE caught fire 4 years ago, and a pallet of batteries that was dropped from a height of 50 feet caught fire ON THE RAMP (not in a plane) 16 years ago. Lithium batteries can still be transported on cargo flights. Otherwise, how do you think iPhones get from China to you? On tramp steamers? They can be carried in the passenger cabin, and they can be contained in portable electronic devices. Because if one does catch fire in the cabin it can be detected and dealt with. Even this has only happened once of the past 10 years. Out of approximately 50,000 flights per day worldwide. That's roughly 183 MILLION flight in 10 years, and one incident.

You will also notice that the battery pictured above DID NOT catch fire. It expanded. As it was designed to do at end of life. Apple makes safe batteries. Putting a "brand X" battery in your iPhone does not guarantee that it will be safe, however. And the battery that expanded was badly abused; allowed to remain in a discharged state for "3-4 months" per the OP.