Imagine a flat earth but with a gravity well at the center. Would its gravity be the same like ours?
Not at all. Gravity would act directly towards the gravity well - it might be fine in the centre of the plane, but as you moved towards the edges the gravity well would no longer be below you, but behind you. You'd feel like you were going up a steeper and steeper slope, as if you were in a huge bowl.
It would work better if the gravity well was some distance below the plane - I'd say at least the same distance from it as the width of the plane itself.
The surface would eventually conform until there is no gradient along the surface. In other words, the shape of the world would change until gravity was everywhere perpendicular to the surface. This might be violated for relatively small structures like hills or mountain ranges, but across the world, it would be a clear effect. So if the center of gravitation is not very far away, the surface would not remain flat.
The strength of a gravity field depends directly upon how much mass is adjacent to the observer. Observe, for example this map of our Earth's gravity field by NASA:
The earth in fact has differing strengths of gravity fields depending on what exactly the crust underneath is composed of. Notice the giant low-grav zone in Canada, a well known phenomena, due to the nature of the crust underneath.
Also, note this older article from National Geographic, they mention that the gravity fields change over time (winds and currents move water, which has enough mass to impact gravity): National Geographic Gravity Map Article
Thus a flat and/or disk shaped world would in fact produce normal gravity, IF the mass was relatively evenly distributed across it's surface, and was fairly uniform in thickness. However, towards the edges, and near mountains or oceans, it would get a little weird.
Using a "gravity well" of some shape would also affect things, if you weren't relying simply upon the sheer mass of the flat earth to produce the gravity.