TX6/FM1960 used to be a country highway. It is heavily developed today and runs through highly commercialized areas. You'll find that the segment from I-45 west and south to the Southwest Freeway (US59S) is multiple lanes with heavy traffic and stoplights every 3-4 blocks.
The easiest way (though not the cheapest) around Houston is the Beltway 8/Sam Houston Tollway option. Next is I-610 (loop). Third is just batten down the hatches and go straight through town. In any case, it's much easier if you time your trip through Houston for off-peak hours.