It looks like a flowering quince. They have such beautiful flowers but are hard to keep nice form. Yours could probably use some more pruning (in late spring or early summer after flowering). It is recommended to prune to 6 -8 long, tall branches removing any crossing or badly spaced stems. To increase flowers for next year - spur-prune - or cut back sideshoots to two or three leaves - to produce more abundant flowers. It is probably best to renovate your bush over two or three years. Good luck,
nancy

That was definitely it. I thought I had seen something similar in a catalog a while back but couldn't put my finger on it. I wonder if it will develop fruit? Now I just have to get rid of the mulberry that is invading the quince's territory.

Thanks nancy0903 for the help!

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The quince that produces fruit and this one are two entirely different species - no fruit on this one.

My flowering quince 'Texas Scarlet' produces fruit.

Yes, flowering quince often fruit and these can be used as well as those of Cydonia oblonga.

yep, needs a major but careful haircut.