Most job interviews are mind games that are not designed to appeal to or be understood by someone who is autistic. I quickly learned you have to fake it until you make it. I learned what NT's look for and expect in interviews at a very young age and since then, I have been hired every time ON THE SPOT during a first interview. My most recent job hiring experience marked a new achievement. I didn't even have an interview. I simply walked in knowing they were hiring, did my thing, and the General Manager was so confident in me that she took me to the back, had me fill out paperwork, and I proceeded to work a six hour shift that same day.
Interviews are very draining, even for NT people, and being autistic makes it ten times worse. Sometimes it feels like you're trying to navigate through an impossible maze. It's very likely that despite your talent, they didn't see the outstanding confidence most people want to see during an interview. I've noticed that many places will hire the most confident individual even if they don't have as much training or education as another applicant. NTs will sometimes avoid hiring someone who comes off as awkward, with no sense of self, quiet or aloof, and who seems to have no direction or idea of what they want (even if that's not the case). I am an incredibly ambitious person but my natural behavior comes off as odd, questionable, and strange. That's where faking it comes in.
Walk in with your head up, force eye contact (this sucks but it's very important. In one interview this was stated afterwards: "I noticed how you maintain very strong eye contact. That's something I look for"), shake their hand strongly AND initiate the handshake, use confident and enthusiastic words, and always have a plan of what you're going to say before you go in to the interview.
Rehearse how to describe yourself when they ask, come up with a response beforehand so when they ask you about what you expect from the job and where you want to go you can answer without seeming doubtful or caught off guard. Plan everything out ahead of time. They'll always have a question you couldn't prepare for, but having the main / repeated questions down in interviews will make things easier and less tiring. I have found doing this and rehearsing how I want the situation to go in my head many times before the interview made it play out more along the lines of how I want it to in person. A self-fulfilling prophecy I guess.
If you have to make things up about yourself to make yourself seem like a more successful, full person... do it.
Also: don't mention you're autistic during the interview. I hate that it's so stigmatized, but if you feel the need to tell them, do it after you're hired and even after a couple shifts or projects that show you're skillful and a good worker.
It's all just a game, really, and when you think of it that way it makes it easier.
Web design and graphic design is a super competitive market anyway... as you acknowledged, and there's probably an applicant out there who's as talented as you, but with that (real or faked) confidence and persona they're looking for.
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