The question asserts

• it can't be the case that missions have been postponed so that another mission came first either
• The Shuttle missions weren't postponed that much that another mission came before
• Following the Columbia tragedy, they made it so that if a Shuttle mission was postponed all subsequent missions were postponed too

Let's judge the validity of these assertions by examining the schedule history of a few missions.

STS-114

• Baselined OV-104 Atlantis as ULF-1 Crew Rotation flight with launch date of 1/16/03 on 12/6/01
• Postponed launch date to NET 3/1/03 on 9/16/02. Postponement caused by Engine Flowliner cracks.
• Subsequent postponements after STS-107 Accident to NET 7/21/03, NET 10/1/03, NET 12/18/03, NET 3/11/04, NET 9/12/04
• Postponed launch date to NET 3/6/05 on 3/22/04. Changed flight to ISS Logistics Flight LF-1, canceled crew rotation, and changed orbiters to Discovery OV-103.
• Postponed launch date to NET 5/12/05, 5/15/05, 5/22/05, 7/13/05
• Rolled back from pad 39B to VAB on 5/26/05
• Rolled out to Pad 39B on 06/15/05 and set launch date of 07/13/05 on 05/22/05.

STS-121

• Baselined OV-103 launch date of 11/15/04 on 10/26/03
• Postponed launch date to NET 5/5/05 on 3/26/04. Slip due to Columbia accident
• Postponed launch date to NET 7/10/05 on 10/29/04. Slip due to Columbia accident
• Postponed launch date to NET 7/12/05 on 2/17/05 to provide on acceptable launch lighting conditions
• Postponed launch date to NET 9/9/05 on 5/23/05 to reflect latest planning decisions
• Postponed launch date to TBD on 11/15/05
• Postponed launch date to 5/10/06 on 3/16/06
• Postponed launch date to 7/1/06

STS-115

• Baselined OV-104 launch date of 4/10/03 on 3/7/02
• Postponed launch date to 5/23/03 on 10/8/02; delays due to engine crack repairs
• Postponed launch date to NET 8/21/03 on 3/13/03
• Postponed launch date to NET 10/30/03 on 4/17/03
• Postponed launch date to NET 1/22/04 on 5/28/03
• Postponed launch date to NET 7/24/04 on 7/29/03
• Postponed launch date to NET 2/10/05 on 10/3/03
• Postponed launch date to NET 8/28/05 on 3/22/04
• Postponed launch date to NET 12/8/05 on 10/29/04
• Postponed launch date to NET 2/16/06 on 5/23/04
• Postponed launch date to NET 7/1/06 on 10/31/05
• Changed launch date to TBD on 11/15/05
• Changed launch date to NET 8/28/06 on 3/16/06
• Changed launch date to NET 8/28/06 on 3/16/06
• Advanced launch to 8/27/06 on 8/3/06 (actual launch date was 9/9/06)

STS-125

• Added STS-125 to FDRD - launch date of 08/07/08 on 06/29/07.

• Ppd. to 08/28/08 on 02/14/08. Slip due to ECO sensor problems experienced during December launch attempt of STS-122.

• Ppd. to 10/08/08 on 05/27/08. Slip due to delays in delivery of ET 127 & ET-129 (STS-400).

• Ppd. to 10/10/08 on 09/08/08. Slip due to Hurricane Faye impacts to HST payload readiness.

• Ppd. to 10/14/08 on 09/24/08. Slip due primarily to training time lost in the aftermath of Hurricane Ike.

• Ppd. to NET Mid-Feb 2009 on 10/02/08. Slip due to HST on-orbit failure of A-side of Control Unit Science Data Formatter.

• Ppd. to NET Mid-May 2009 on 10/30/08. Slip due to checkout problems with HST spare control unit.

• Selected May 12, 2009 launch date on 12/04/08.

• Advanced from 05/12/09 to 05/11/09 on 05/01/09. Advancing one day provided a 3rd launch opportunity before range conflicts.

Data from Space Shuttle Missions Summary.

I urge you to read the POSTPONEMENTS block for each mission you are interested in. Postponements are changes to the scheduled launch date, scrubs are slips after the scheduled launch date.

Other questions of interest about shuttle launch scheduling and nomenclature:

Why so many STS-61's?

How often were Space Shuttle launches scrubbed?

¹ Before and after the goofy "let's hide our launch scheduling issues by obfuscation" era, that is

The Shuttles and their payloads were assigned their flight designations when the mission was planned, not when the mission would be launched.

Interestingly enough, NASA considered at one point changing the designations as the launch order changed, but officials quickly realized that this would have confused too much of the mission paperwork. Once the flight designation was assigned, it stuck. No more changes could be made. That way, no matter what delays or reschedulings moved the launch around, the designation for a mission would be the same—from the day it was first planned to the day it landed.

From Behind the Space Shuttle Mission Numbering System

¹ Before and after the goofy "let's hide our launch scheduling issues by obfuscation" era, that is. See Why so many STS-61's? for more info

Data from Space Shuttle Missions Summary.

STS-125

• Added STS-125 to FDRD - launch date of 08/07/08 on 06/29/07.

• Ppd. to 08/28/08 on 02/14/08. Slip due to ECO sensor problems experienced during December launch attempt of STS-122.

• Ppd. to 10/08/08 on 05/27/08. Slip due to delays in delivery of ET 127 & ET-129 (STS-400).

• Ppd. to 10/10/08 on 09/08/08. Slip due to Hurricane Faye impacts to HST payload readiness.

• Ppd. to 10/14/08 on 09/24/08. Slip due primarily to training time lost in the aftermath of Hurricane Ike.

• Ppd. to NET Mid-Feb 2009 on 10/02/08. Slip due to HST on-orbit failure of A-side of Control Unit Science Data Formatter.

• Ppd. to NET Mid-May 2009 on 10/30/08. Slip due to checkout problems with HST spare control unit.

• Selected May 12, 2009 launch date on 12/04/08.

• Advanced from 05/12/09 to 05/11/09 on 05/01/09. Advancing one day provided a 3rd launch opportunity before range conflicts.

Data from Space Shuttle Missions Summary.