Western states have a lot of Fed and State land.Much of it allows shooting.
You DID qualify your question with "safely"...I noticed and appreciate it.
But,to emphasize:
Public land is multi use,and there is no secure downrange beaten zone,in other words,you don't know where another person migh also be enjoying the public land.They may be on a horse,bicycle,or on foot.They may not be visible,but "grazing fire"over the horizon may affect them.
Public land often has grazing leases,which means livestock may be anywhere,and the folks checking stock or fences or gates could be antwhere.
Public land is often checkerboarded with private land,so within public lands will be homes,ranches,people.
My point being,when you say "run and gun",its not good enough that you do not see anyone in the direction you are shooting.You must know where each round stops in a backstop.
In my area,there is a National Grassland where shooting is permitted.It is also true residents ,ranchers,and other folks in the area have had issues with stray bullets endangering their lives.
They are not happy about it,it gets in the news,and they ask for political action.
So,to answer your question,are there large trats of public land where a person can shoot?Yes,but check for restrictions.
Are there areas on public land that are free fire zones for run and gun?(safely of course)...Well,that "safely" thing has to be carefully considered.
All of the general public has access to all of the land surrounding you,and they may access it from a different direction you do not know about.
Generally,I suggest you will not find many places on public land that meet your needs.
If you get any complaints,the Ranger or Officer may well be inclined to write you up for something like felony reckless endangerment with a firearm.They do take these things seriously.
I will also suggest,at least in Colorado,they merged the Park administration with the Division of Wildlife.
That means,while in the past the officer who came to check on folks shooting on public lands was typically a game warden.The DOW officer would be comfortable with hunting and guns.
Now it will be a Parks and Wildlife officer.Its a whole different profile of a person.Some figure open land is for birdwatchers and lycra shorts bicycle riders and they don't like guns,hunting,or shooting.
There are programs where tax dollars are used to aquire large tracts of "open space" in Colorado.The mentality of the "Parks"type administration shows up here,as these tracts typically do not permit shooting or hunting.
This is likely to be the ideology of the of the ranger who climbs out of the pickup to check you out.
