It's more than just a little heavy-handed. It's a LOT heavy-handed.
SOPA and PIPA are designed to give the government more tools to deal with sites that host and distribute copyrighted content outside of the US (read: Pirate Bay). Under the current laws, they are outside of the gov's jurisdiction. So instead of being able to shut them down
directly, SOPA and PIPA aim to prevent distribution through a few methods:
1) They can go after search engines like Google and Bing to have infringing sites scrubbed from their indexes.
2) They can go after other sites in the US that link to other sites hosting or distributing infringing content.
3) They can go after ad networks that run ads for infringing sites and have their ads removed, thus removing any possible ad revenue.
4) They can go after payment processors like PayPal and credit card companies to shut down merchant accounts of infringing sites, thus cutting off donations and payments they might receive.
5) Finally, they can go after ISPs and force them to remove or change DNS records for infringing sites, which would make them harder to get to.
The last part is the scariest, because it would potentially give the government control over one of the most critical pieces of the Internet infrastructure. DNS, in a nutshell, is a directory of names to IP addresses. Every site on the Internet has a numerical IP address, this site's for instance is 64.18.100.104. DNS takes an easy-to-remember name, such as
www.volksfolks.org, and translates it into the IP address. The DNS service is run by ISPs and other internet companies and it's designed to distribute all the names and addresses across the internet. As you can imagine, if the government was able to order all of these ISPs to remove a DNS record for an infringing site, it would effectively be inaccessible (except by its numerical IP address).
Last I heard, the DNS portions of the bill were off the table. However, with all the other stuff it allows, it's still a very dangerous bill.
Under the current copyright laws (The DMCA which was passed in 1998), here is how a copyright claim is handled. Let's say me or anyone else here posts some kind of copyrighted content to the site. It could be a song, photograph, movie, whatever. The copyright owner finds this and wants it removed. What they would do is send a letter to our network provider requesting a takedown of the infringing content. The provider would have to respond by either asking us to remove the content, or they could shut us down completely if we don't. Either way, the provider has to reasonably comply with the request, in order to keep "safe harbor" protection. That means a network provider isn't liable for any acts of copyright infringement its users commit (as long as they respond to DMCA takedown requests).
Obviously, the DMCA only applies to networks within the US. But let's see what would happen if a bill similar to SOPA/PIPA passes into law. The intent of course is to go after copyright infringement sites outside of the US, but it most certainly could apply to sites within the US as well. Let's say the same thing happens and one of us posts some copyrighted content to our site. Or not even that, let's say someone just links to a site from here. The law is so vaguely written that sites could be in violation if they "engage in, enable, or facilitate" copyright infringement. Is linking to a site hosting copyrighted content "facilitating" it? Someone could convince a judge somewhere of that I'm sure if they wanted to. And thus, our site could be taken down and found liable of "facilitating" copyright infringement just for linking somewhere.
That's kind of an extreme example, but with such vague wording and flexibility, these laws would almost certainly be misused, much like the DMCA is misused now in a lot of situations. Any site that allows comments or users to post material to them would be in danger of "engaging in, enabling, or facilitating" copyright infringement.
That's a pretty scary situation to be in, and yes it would put us a few steps closer to countries like China in which the government monitors and controls Internet access and usage. That's why a lot of sites went down in protest of these bills, and also to raise awareness. Because even if SOPA/PIPA don't pass (and right now it doesn't seem likely in their current form), other bills will almost certainly be proposed that do the same thing. As you can imagine, the co-sponsors of the bill received a tremendous amount of contributions from the entertainment industry, who of course wants this to pass. They have powerful lobbyists with deep pockets.
No one is arguing copyright owners shouldn't be able to protect their content from being pirated, but this is clearly the wrong way to go about dealing with piracy. In fact not only would it not prevent piracy, it would deal a huge blow to the free and open Internet we all use today.