My front turn signals stay on as running lights. There two filaments in the two bulbs, one that runs all the time and one that blinks for a turn signal.
Here is a picture of the bulb to get an idea of how it works.
The two prong one is the running light and the three prong one blinks for a turn signal.
I would take off the lens and see if the two prong one is broken. If it is, then its your bulb. Get a couple of replacements and it should be good to go.