Welcome to ME!
Yep, sounds like factory suspension to me. Very bouncy and hard in the front.
For a tighter suspension, you can upgrade your springs, shocks, and struts. I'd do all three at once and try to get all three components from the same manufacturer (i.e. Steeda, Eibach, Ford Racing, etc.). This can eliminate nose dive from hard breaking, rear-end squat from hard acceleration, definitely stiffen up the ride, and usually level out and lower the car for a nice look.
To reduce body-roll when making hard turns, you can replace the front and rear sway bars along with a strut tower brace under the hood.
For more info, AmericanMuscle has a few write-ups themselves (if you care to read):
Scroll to the bottom for advice section: 2010-2014 Mustang Suspension*|*AmericanMuscle.com - Free Shipping!