Havant & Waterlooville beat Aldershot 2-1 live on BT Sport on Wednesday night to claim their first home victory.
It’s been a mixed first nine games for the Hawks but the National League South champions feel they are now up and running.
Bradbury is a realist and says ambitious are greater than just finishing fifth from bottom.
“As long as we are in the league next season that’s great, but we want to do a little more than that,” he told the league’s Oli Osborn.
“We want to build. With the group of players we’ve got who work very hard for each that’s possible.
“This league is relentless. In the National League South if you make a couple of mistakes you can get away with it, but you are playing against better players and better managers who have all had better coaching.
“It’s where we want to be at this level. We wanted to play with no fear in front of the cameras and we showed that we deserve to be in this league.”
The win was gritty, and Bradbury says the rub of the green was deserved after some poor fortune in games so far.
“We scored a couple of goals off the training ground and that was nice,” he added. “We’ve been due a bit of luck and it came our way with one of the goals.
“We were strong and organised - we knew we would have periods without the ball but we kept to our game plan and gained a valuable three points.”