After which battle did both sides realize that this would be a long and bloody war?

The bull run

Explanation:

The First Battle of Bull Run, was also known as the Battle of Manassas,it was a major land battle of the American Civil War and also the first. Union and Confederate armies clashed near Manassas Junction On July 21, 1861 Virginia.

The fight started when about 35,000 Union troops marched from Washington, D.C. to strike a Confederate force of 20,000 by the Bull run, a small river. After fighting on the defensive for most of the day, the rebels broke through and sent the Federals side back towards Washington.

This victory by the Confederate gave the South a lot of confidence and left the north shocked upon the realization that the won would not be easily won as they had thought.