No, they don't have sufficient thrust when they're jettisoned to catch up with Space Shuttle accelerating away:
The SRBs are jettisoned from the space shuttle at high altitude, about
146,000 ft (45 km). SRB separation is initiated when the three solid
rocket motor chamber pressure transducers are processed in the
redundancy management middle value select and the head-end chamber
pressure of both SRBs is less than or equal to 50 psi (340 kPa). A
backup cue is the time elapsed from booster ignition.
The separation sequence is initiated, commanding the thrust vector
control actuators to the null position and putting the main propulsion
system into a second-stage configuration (0.8 second from sequence
initialization), which ensures the thrust of each SRB is less than
100,000 lbf (440 kN). Orbiter yaw attitude is held for four seconds,
and SRB thrust drops to less than 60,000 lbf (270 kN).
At the same time, Space Shuttle continues thrusting with its three SSME, each at roughly 418,000 lbf (1,860 kN) for 5,250 kN (1,180,000 lbf) total thrust to the remaining mass of the launch vehicle.
So if we assume worst case, SRB dry mass is 82,879 kg, and at maximum separation thrust can achieve acceleration of 3.26 m/s² which will rapidly decrease as residual solid propellants burn to exhaustion, while the remaining Space Shuttle Orbiter + External Tank configuration weighs approximately 874,290 kg when full (it wouldn't be even close to that on SRB separation) for a minimum acceleration of 6 m/s². So even assuming absolute worst case, thrust of Space Shuttle SRBs on jettison isn't enough for them to catch up with the Space Shuttle.
I used numbers from Braeunig.