You might want to check with your local government about permits to build over a stream, and you most likely will need to get written permission from the Army Corp of Engineers.
While state laws vary, federal law doesn't. Our church plans to build some bridges, 4 small ones, over one decent size creek and two spring branches. We have had to get permission from the Corp, get plans approved by the county, and at that, the bridges have to have the abutments 3 feet back from the bank on each side to meet requirements.
Most will scoff and say that it isn't necessary or worth the trouble to go through all of that for a small bridge. Just do it and don't worry about it.
We have a small horse farm, boarding enough horses to keep hay in the barn and the lights on. In other words, there are enough borders to pay expenses, but no profit. That enables my daughter to keep her horses without incurring so many personal expenses. I say all of that to say that the operation is a very small one.
We had a private environmental group, which drives around and looks for farms that operate with creeks, streams, and ponds involved, and turns them in if they believe that there are infractions. We went through a year's worth of headache, and about $15,000 worth of changes, to comply to the agriculture department's requests (manure pile too close to creek and erosion of banks due to horses entering and leaving two ditches). They were very reasonable, but as they expressed, they were stuck between reasonable practices and nosy environmental groups. It happens, these people who anonymously stick their noses into other's business, and then disappear while you try and comply.
I'm not saying that you should go through all of those hoops- I think that it is ridiculous. But, if this bridge is remotely visible from the road, do not discount the possibility that someone, someday, will turn you in. And, what we went through pales in comparison to the fines and trouble for improperly building a bridge over a creek, or ignoring the Corp. Our church was warned that if we built those bridges (we had gone through some other processes to run water and sewer under the same creeks, legally, and the bridges came up in the conversation) without complying, we would be facing fines of $10,000 per bridge.
It stinks, but we live in an age where your property is not yours to do as you please. The government, and private environmental groups, have more power over your land than you do.