Narrative:

My student and I were in [a small aircraft] on right-midfield downwind for runway 13. Another training aircraft was final for 13 doing a touch-and-go. Tower was going to try and get another aircraft out between me landing (touch-and-go) and the other aircraft departing. My student and I received instructions to extend our downwind leg and tower would call our base. The airplane cleared for departure was slow moving and tower assigned my student (who was flying at the time) to enter a right base for runway 18; we were not cleared to land. I watched the aircraft on runway 13 begin to line up on the center line. The situation did not feel right; so I took the controls and slowed the aircraft from the typical 70 KIAS used on a base leg to 60 KIAS to try and create more time for the aircraft departing runway 13. I then called tower to get approval for a left 270-degree turn back to final. I had no response. I began making s-turns on final to; again; create more space. I heard tower instruct a [twin aircraft]; who was on an instrument approach; that they were not cleared for 13; but to make a right base for 18. Tower then told me to make a left 270 to final for the touch-and-go. I stated that I was going to depart to the east and come back when things had calmed down. Tower instructed that I make a left 270 to enter on final for 18. I started making the assigned turn and was not able to see west due to the sun. As the turn came around through north; the [twin aircraft] was wing up turning final for 18. We were no farther than 30 yards apart. I yelled over tower frequency for the [twin aircraft] on final to go-around and increased my bank hard to the left. Tower then instructed the twin to go-around. The twin stated that they were going to make this a full-stop and landed on runway 18. Tower then instructed me to make another left 270-degree turn and was cleared touch-and-go for runway 18 and asked if I had the [twin aircraft] in sight. I said that this would be a full stop and said that I did not; assuming that there might be another [twin aircraft] in my area. I was cleared to land runway 18 and made a wide turn; looking for another suspected [twin aircraft]. I landed and taxied clear of the runway as instructed.

Related reports:

Google
 

Original NASA ASRS Text

Title: Pilot providing instruction at PVU described a near miss when ATC's instructions resulted in a conflict with other pattern traffic.

Narrative: My student and I were in [a small aircraft] on right-midfield downwind for Runway 13. Another training aircraft was final for 13 doing a touch-and-go. Tower was going to try and get another aircraft out between me landing (touch-and-go) and the other aircraft departing. My student and I received instructions to extend our downwind leg and Tower would call our base. The airplane cleared for departure was slow moving and Tower assigned my student (who was flying at the time) to enter a right base for Runway 18; we were not cleared to land. I watched the aircraft on Runway 13 begin to line up on the center line. The situation did not feel right; so I took the controls and slowed the aircraft from the typical 70 KIAS used on a base leg to 60 KIAS to try and create more time for the aircraft departing Runway 13. I then called Tower to get approval for a left 270-degree turn back to final. I had no response. I began making S-turns on final to; again; create more space. I heard Tower instruct a [twin aircraft]; who was on an instrument approach; that they were not cleared for 13; but to make a right base for 18. Tower then told me to make a left 270 to final for the touch-and-go. I stated that I was going to depart to the east and come back when things had calmed down. Tower instructed that I make a left 270 to enter on final for 18. I started making the assigned turn and was not able to see west due to the sun. As the turn came around through north; the [twin aircraft] was wing up turning final for 18. We were no farther than 30 yards apart. I yelled over Tower frequency for the [twin aircraft] on final to go-around and increased my bank hard to the left. Tower then instructed the twin to go-around. The twin stated that they were going to make this a full-stop and landed on Runway 18. Tower then instructed me to make another left 270-degree turn and was cleared touch-and-go for Runway 18 and asked if I had the [twin aircraft] in sight. I said that this would be a full stop and said that I did not; assuming that there might be another [twin aircraft] in my area. I was cleared to land Runway 18 and made a wide turn; looking for another suspected [twin aircraft]. I landed and taxied clear of the runway as instructed.

Data retrieved from NASA's ASRS site as of July 2013 and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.