The first happened in the underground section of the car park near Asda at around 6.30pm on Sunday (21/02) when a group of around 15 to 20 teenagers were involved in an altercation.
Nobody was injured but a scooter was stolen.
A short while later there was a second disorder close to Trowbridge Railway Station where a car was damaged.
Officers attended and made two arrests.
A 16-year-old boy was arrested on suspicion of affray and criminal damage, and a 14-year-old boy was arrested on suspicion of affray and theft. They have both been released on conditional bail.
PC Jordan Gratton, from the Trowbridge Area Community Policing Team, said: “Although we have already made two arrests, we are keen to speak to witnesses who saw what happened on Sunday.
“We know that both of these disorders would have been witnessed by passers-by, including motorists who may have dash cam footage of the two incidents.”
If you have any information or can help in any way then please call us on 101, quoting crime reference 54210017166. Alternatively you can call Crimestoppers anonymously on 0800 555111.