Where can I publish my work online and get paid for it?
I'm just new to this and this is my first work and it's still chapter 1. It's a fantasy and mythical story.
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Short answer: You can't.
It takes time to create longer works. Even experienced writers will go back and edit earlier chapters, move things around, etc. If you're not experienced, you may do this even more as you learn your craft.
To publish a novel, including self-publishing, it needs to be complete. You might only put out a chapter at a time, but all the chapters are ready to go. No publisher will take on a novel at any other stage, unless you're very famous.
An exception is a series. For example, my spouse has a 5-year contract with a publisher for his comic series. It comes out quarterly (in theory). In order to get the contract, he had the script for the first year complete, plus some of the art. With most of the script done for the 2nd and 3rd years, and outlines for the rest.
To make money with your writing, start small. Find sites that pay for articles or short stories and submit to there. Some nonfiction sites will assign you work, others will consider already completed work.
Don't expect a lot of money. If you have to pay someone else to complete your work (an artist, for example), publishing is often about losing money. You will get paid outright for one-offs in a magazine or some anthologies (assuming you get paid at all) but mostly any payment is through royalties. Which means you may never see a dime. Read your contracts carefully but even good contracts don't mean you'll make money.
More experienced writers can put ebooks on Amazon and other sites, offer work on Patreon, submit comics to ComiXology and DriveThru Comics, etc. If you have a great idea, you might be able to use Kickstarter, Indiegogo, and so on.
Bottom line: keep writing, but get a day job.
If you want to publish a shorter work AND get paid for it, then it almost certainly needs to be a complete stand-alone short story, not the first chapter of a longer work. If you can successfully rewrite your chapter into a stand-alone story, then there are actually plenty of both print and online outlets that buy stories in the fantasy genre. The most prestigious and well-paying are well-established print publications like The Magazine of Science Fiction and Fantasy. Here's an additional list of other paying fantasy markets.
It would be irresponsible to not warn that fiction writing is most definitely NOT a reliable source of quick money, particularly for a new writer. There is a LOT of competition, and many of your competitors will have been writing professionally for years and years. Even well-established writers often struggle to pay the bills, or work second jobs to subsidize their writing careers. It's quite common for writers to have years of writing under their belts before their very first paid sales. You can always self-publish online, and put your work up for sale, but it's very unlikely that people will pay for an unfinished book by an completely unknown author (unless you're an amazing salesperson).
If you just want to be read, and you don't care about being paid, there are a lot of non-paying online outlets that feature work by all comers. In particular, if you write fan fiction, or work that plausibly fits into the world of an established fantasy franchise, there are passionate communities of fans who might embrace your work, and established places to publish it. In that case, however, you're basically legally prohibited from profiting from your writing (the profits would belong to the copyright holders of the franchise).
I have seen authors with Patreons who mostly sate their patrons with chapters posted as they write their stories. Then when the story is completed they put it up for sale online publicly (possibly provided for free to their patrons).
But in every case, these authors have built up a following by posting their work on free sites for a significant period of time. Patreon is not a site structured around paying for content - it's structured around supporting creators whose content you are already consuming. Patreon does feature exclusive rewards which create a "pay for content" experience, but trying to post your content exclusively on Patreon and charge for it is doomed to failure, because it misunderstands the motivations of the patrons.
And even if you did manage to make such a business model work, you're still going to making maybe a couple hundred a month. Enough for a trip to Disneyland (assuming you're going alone) or a really nice blowdryer, but not enough to live on.*
There's no "turn writing into cash" button - it requires work and dedication (and
a little bit a great deal of luck) for all authors.
*I write this, knowing that you go to Seanan McGuire's (the author telling the blowdryer story I linked to) Patreon page, you'll see that she makes several thousand on Patreon each month. But she's also a Hugo and Nebula award-winning professional author with several ongoing and completed series to her name. Her success on Patreon was earned from her work outside Patreon, and should not be taken as a measure of achievablity by a new author.
So, while I am in general agreement with the other answers that are saying "You Can't", let's see if we can create a situation where "You can."
First, let's take a look at what your work is. Oh, it's a fantasy? Well, good for you that Fantasy is currently one of the most popular genres amongst young adult readers in many parts of the world. Let me explain: In Japan, China, and South Korea, fantasy stories have become one of the most popular genres in part thanks to the wild success of stories such as Sword Art Online (a sci-fi disguised as a fantasy) and the omnipresent Disney's romanticization of traditional fairy tales which include many of our traditional fantasy elements. In the West, we have fantasies like Lord of the Rings, Game of Thrones, Star Trek (space fantasy), Percy Jackson (urban fantasy), and The Hunger Games (sci-fi fantasy) alongside a bountiful trove of myths and legends like the stories of Charlemagne and Arthur which permeate ubiquitously through our cultures. The younger the audience, the more attractive these fantasy worlds become as an escape from our dreary real-world. It's these 18-28 year old consumers who would be most readily willing to give money.
But wait, you only have a single chapter written. You can't get published with only one chapter... or can you? Websites such as Wattpad and Tapas are probably your best bets for serialization if you want to spread your story quickly and effectively. Additionally, reading your story on YouTube and sharing [Read-Only] Google Doc versions of it on Twitter can be great resources as well for getting your content to people who already follow you and have a specific interest in what you have to offer.
But these resources only distribute your work. They won't result in you getting paid for your writing, will they? That's where Patreon comes in. After you have uploaded a few chapters publicly, make a Patreon account as a writer. You can distribute your writing through Patreon as well, but be careful to not act like it's all about the money because it shouldn't be. You still need to distribute your story for free. Keep the serialization going publicly. Let people read it for free. At the same time, by having a Patreon open, you can encourage those who want to help support your writing to donate so that you can gradually make writing your main and then only job. Once you have finished a volume of your story, make an Inkitt and a Swoonreads. Share that finished volume to these sites where you can potentially gain an even larger following who might be willing to support you. Additionally, by sharing your story publicly, if you do well-enough, your story can get picked up for publication, meaning you can earn money from that. Alternatively, you can also self-publish your finished volume online through companies such as Lulu, Amazon, and Barnes & Nobles where you can sell your book through their individual marketplaces! All the while, by having active accounts on all the preceding websites, you can alert your readers that "Hey, if you want to support my writing, you can buy Volume 1 on [insert website here]!"
In short, if you're trying to make a lot of money fast on a chapter-by-chapter serialization, don't waste your time. It's better to build a fan-base who is actively looking forward to your next release and then move towards monetization after deciding if it is right for you and your story in the moment. By working hard, you gain notoriety and respect which can be used to make it easier for people to want to support you. In time, you can eventually start making money, little by little. If any company is interested in publishing you through their imprints, then they will extend an offer to you. Remember, you should write for the love of writing. If you're doing it to make money and not for the joy of writing, you should stop because this is not the right job for you and it will stress you out. Take the time to write for fun, not profit. Let opportunities to make money from it find you.
Novel one chapter at a time, or one part at a time on Amazon Kindle in their Vella program. First two chapters are free to the reader, presumably so they can see if they wanna read the whole thing, then they pay by the chapter after that, as you submit them. Hopefully you don’t take forever to add new chapters thereby boring your reader to death waiting.