I want to start out by saying that I think it's great that you want to do something. Not everyone is motivated to help make positive changes in our world. I think that you need to do some more research, though, and revise your strategy if you truly want it to be successful. A little more effort could make what you want to do truly successful!
I've been riding horses for almost as long as I can remember. I own two 'rescue' horses and have been actively involved in most things related to horses for YEARS.
Unfortunately, motor vehicle accidents aren't the biggest crisis facing carriage horses in NYC. To be honest, what happened WAS a freak accident. This is NOT at all common among the horses in NYC. Their living situations are not always ideal - or close to it. Admittedly, the horses are in much better shape (and seem to be kept better) than they were even 5 or 6 years ago due to a lot of press on this issue a few years ago, so things appear to have gotten better for them. (And a lack of shade isn't honestly a big problem for these horses....)
However, if you REALLY want your petition to be useful and taken seriously, it can't just say 'We don't want horses getting hit by cars or treated badly! Do something!' It has to offer a logical and realistic solution and it has to offer more substantial facts and proof about what is actually happening. It should mention (and possibly quote) the report to which the article above refers. It also shouldn't be online. There is very little proof out there that online petitions do anything - b/c there's almost no way to verify who actually 'signed' the petition. Politicians don't normally hold them in high esteem. (Before I was a lawyer, I worked for nonprofit groups doing lobbying work. I worked very closely with local and state representatives, so I do have experience here)
I've been following what has been going on with the carriage horses in NYC for over 10 years. It honestly HAS gotten better. Is it enough? No? Are you ever going to get them banned? NO! So you need to logically (and without bare, angry allegations) propose a workable solution. Note, banning carriage horses would make their situation worse. These are not people who easily afford their horses. They need to work to be able to afford them. They don't view their horses as pets -- We have to recognize that this is fine *IF* they take appropriate care of their horses. Banning carriage horses (and I'm not saying this was your suggestion) would only subject these horses to a worse fate. Few of these drivers can afford (or have the inclination) to send their horses to retirement farms if that were to happen. I'd venture to say even the same amount wouldn't be able to afford euthanasia. Theses horses would wind up at a regional sale... (That's not usually a good thing. Many horses (though admittedly only the lowest priced) at sales wind up being sold to meat men. Despite the fact that we've banned horse slaughter in the US, we don't yet have a law banning the shipment of horse out of the country for slaughter. The killers now simply take the horses to Mexico, where slaughter conditions are worse).
Your heart is in the right place, but an online petition without any factual back up or anything else is not going to make a dent. There have been thousands of them. A letter writing campaign would -- A thoughtful, well written, substantiated form letter could be drafted. People could download the letter, sign it, and mail it to the mayor.
Finally, you need to do a lot of research on exactly what is involved in proper horse care. I am in the city 5-6 days a week. I see these horses almost every day. I haven't seen a sickly looking or underweight one in quite some time. That doesn't mean that their living situations are okay -- but certainly they aren't starving. Since many look okay, you're going to have to substantiate your claims with (1) substantiated FACTS about how horses are being kept and (2) what would be an acceptable level of care GIVEN that they are working horses in an urban environment. You cannot keep horses of their size in appropriate weight (or even remotely decent weight) without feeding and watering them. They are 1,200-1,600 pound horses for the most part, working most of the day. They wouldn't stay that way without some sort of feed. I've also seen more and more people offering horses feed and water while working. And again, the lack of shade thing isn't really appropriate. (Horses outdoors rarely seek shade for long periods of time, though it's good for them to have the option.)
You REALLY need to research exactly what the problems are and exactly what horses really need. They are not dogs and cats. Their diets are extremely different.
Finally, we all have to come to terms with the fact that they are working horses. So long as they are taken care of, we realistically can't ban working horses just b/c they don't live in a big green field and graze all day long. Cowboys still exist in this country, too, and their horses work far harder than the CP carraige horses do, for example. So, the relative inquiry needs to be 'What is an appropriate level of care and safety for a working carriage horse in Manhattan, and what can we do to ensure the horses are kept healthy and safe?' Then, 'What is the level of care and safety currently and how can we raise that level to the appropriate level for this situation?'
I'm happen to answer any questions you may have about horse care -- and please note that not everything you read on the internet about horse care isn't correct. There are a lot of bad horse people out there.
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There's also a lot of good information, but be sure to verify it with a good horseperson or two.
It's GREAT that you want to do something, but with a little more work, you can put something together that has a much better chance of being successful.
Feel free to ask me any questions you may have.
Good luck.