As others have already stated, getting yourself fired on purpose is a bad idea. Most of us work in a fairly specific job and people "in the industry" know each other.
There are ways you could potentially get around the 4 weeks:
-Have you accrued vacation time? Book 4 weeks off and show up on the first day of your holiday to give your 4 weeks' notice.
-Does your prospective new employer know you have a 4 week notice period? It really depends on what the job is, how bad they want you and how hard it is to find someone who's almost as good as you, as well as how high up in the company is the person hiring you. For example, I've been hired directly by the CEO of a very rich, but very small (in terms of employees) company. I've also been hired by the #14 highest up in the HR department of a shitty, but massive conglomerate. Depending on the situation vis a vis, who wants you at the new company and how badly they want you, you may be able to ask them to help you negotiate an early release. Failing that, if you're honest about needing time to get started, is it out of the question that they wait a little bit for you?
-Have you informed your current employer that you're unhappy? Have you informed them you're accepting another position? They may be sympathetic. Then again, you stated that you're unhappy with the work, and with the workplace. Is management part of what you're unhappy with? If so, would they be petty enough to torpedo you if you just spoke with them honestly about your concerns or about your plans? There are a myriad of possibilities that could come from speaking honestly with your boss, and a lot of information we don't have to help you predict what outcomes are most likely. Could or would they intentionally screw you over if they found out you were leaving? Could your current job become the job you want it to be? Is it possible that they'd fire you automatically upon receiving your notice? I once had a really bad day in a management position due to an incompetent sub-manager misplacing (on paper) almost $100k, adding a couple hours to my shift while my gf waited in the parking lot and I wrote my resignation letter right then and there. Turns out it was the unspoken policy of the company to pay out full severance to and fire on the spot any person at my level who put in their notice. I didn't know this beforehand, because they didn't want people taking advantage, but they also didn't want high-level people who weren't going to stay with the company to be on the premises.
-How are your finances, how big is the company you want to go work for and how unique is this position? If for some reason you just can't get out of the job without trickery, the new job won't help you and won't wait for you either, how often does this job come up? Are we talking about going from McDonalds to KFC? Dishwasher to Fry cook? Mailroom to CEO? Is this a once in a lifetime opportunity to get your DREAM JOB or is it something that you could get hired on to next month, next year? If none of the other parties can work around you, can you afford to just quit, do your notice period and take your chances? Will you be effed if you leave this job and the new job falls through or can you afford to wait - unemployed - until you get what you want?
To sum it up (honestly, I'm bad at summarizing, this might be just as long as the above)
First, consider your situation. You know a lot that we don't, so our advice is worth less than you want it to be.
Second, be honest about your situation, but decide who you should be honest with first. The new employer or the old. Who is more likely to help you? If it isn't the new job, I'm not sure why you're switching.
Third, if you're committed, just do it. Be professional, but just go. Just say "I'm out, here's my notice." If they say "ok, see you for the next 20 work days," well that's just how it is. But maybe they'll say "you don't want to be here? K, FU den, gtfo, here yo money is, baibai!" Or some other unprofessional thing, and then you'll know you made the right decision.
Fourth, (chronologically, but actually most important) believe in yourself. You got an offer for this job you want. If they won't wait for you, well screw 'em. Their competitors will hire you.
Fifth and (FINALLY!) finally, grow the fuck up, man: Work sucks. I hate it. All the other people who have given you advice hate it, too. Even if it's your dream job. Even if it's your own business. Even if it's easy AF, it isn't doing what I want to do when I want to do it because I want to do it. We all get to a point in our lives when we realize what we don't like about our jobs isn't about the job but about ourselves. Ask yourself, is this a life lesson that you've learned yet? I mean really think about that. Do you hate your job because it's a bad job or do you hate your job because you haven't yet resigned yourself to the fact that people pay you for catering to their priorities and don't give the slightest fraction of a percentage of a shite what you think or how you feel? Cos that's how life works. I have 26 employees, and remembering having genuinely shitty bosses, I try and make things as good for them as I can, but at the end of the day, they do the job they agreed to do on contract-signing day or they worry about how to pay the bills, and I don't give a half a percent of a crap about which one they choose, because the wage I pay is way way more than fair for the work.
No matter what you choose to do, in the end, be a grown-up about it and let us know how it went.