Power to light fixture is on, light doesn’t turn on

It sounds wrong to me, but I couldn't be sure without actually being there. You should connect the new fixture, in exactly the same way the original fixture was connected.

If there wasn't a grounding conductor in the original circuit, replacing the fixture did not make one magically appear. If a grounding conductor is required, you'll have to install one in accordance with local codes.

EDITED: Now that you've posted a picture... ;-) Still test to confirm this, but it looks to me as if:

• The 2 blacks and 1 white that are connected, are feeding power through the light box to downstream outlets and also sending power to the switch (on the white wire). Wrap the end of this white wire with black tape or mark it with a black permanent marker. It's a hot wire.

• The lone black wire with a wirenut on it is the power coming back from the switch. This one is hot when the switch is on, dead when the switch is off. Connect this to the black/hot lead on your light fixture.

• The 2 white wires are the neutral (grounded conductor) from the incoming circuit, and the neutral serving the downstream outlets. Connect these to the white/neutral lead on your light fixture. I'd probabaly add a short pigtail so there's only one connection to the neutral on the light fixture, and also so it's that much more obvious which wire(s) connect to the fixture.