(Most recently updated 10/15/2019.) At 568 miles*, Texas 16 is the longest state highway in Texas (that is, the longest road with a state highway designation, as opposed to a US route or an interstate). The second-longest is Texas 6, at 551 miles*. These two routes intersect in the town of De Leon. Actually, they overlap for about a half-mile:
So in 1935, the highway junction in De Leon involved TX 67 and TX 81. The 1939 map was the last that still showed that arrangement:
In 1945, the US 67/TX 67 junction in Dublin was eliminated: the TX 6 designation was extended northwest from Waco, replacing TX 67 all the way to Breckenridge, and then continuing almost to Throckmorton. This increased the total length of TX 6 to over 420 miles, causing it to surpass TX 36 (at 392 miles) as the longest state highway.
That also became the first time that TX 6 and TX 16 shared a junction, but at the time TX 16 was the third-longest state highway. It became the longest 20 years later: in 1965, TX 16 was extended south from San Antonio to its current terminus in Zapata, increasing its length to over 500 miles. After another four years it was extended north to its existing terminus near Windthorst, thus attaining its current length.
In 1975, TX 6 was extended north to its current terminus at the Oklahoma line (along what had been TX 283, which was eliminated because of... yep, you guessed it: a conflict with the nearby US route of the same number).
However, it should be noted that TXDoT did not always eliminate intersections of the same number, and in at least one instance it created a new conflict. In 1947, TX 18 was eliminated in favor of TX 70, which was extended north through Matador, where it intersected US 70 (an arrangement that still exists today).
*These mileages do not match the figures listed on the corresponding Wikipedia pages. I did the measurements manually, and I believe Wiki is in error. I suspect the TXDoT source cited by Wiki editors does not take all concurrencies into account.