Air carrier First Officer reported PGD Tower's failure to provide separation on departure resulted in the flight crew needing to avoid another aircraft; and expressed concern about Tower procedures.

Narrative

I'll start by saying that due to our altitude the TCAS aural warnings were inhibited. Departing Runway 04. Upon rotation and the gear up call; as I reached for the gear handle I noticed the traffic target ahead and just to the right go amber in color. Our clearance was to turn left heading 020 and as we climbed through about 350 feet the Tower reminded us of the 020 heading and to contact Departure. The Captain called pulled heading" and initiated the turn. I was looking out the forward window looking for the traffic as I pulled the heading bug. As we turned; I saw a GA aircraft pass above and just to our right through the forward window. I don't know if this aircraft was the previous aircraft to land and was on a touch and go remaining in the pattern in right traffic or if he was just entering the pattern. I replied to the Tower that I had the traffic in sight and over to Departure.In our critical phase of flight to have another aircraft operating so close is dangerous and it is a continuous problem in Punta Gorda. It feels as if the controllers are comfortable with things always going right. Had we encountered a bird strike; an engine failure; anything that may have distracted or thrown us off we may not have initiated that turn so quickly or even became aware of the other aircraft. It is hard to say how close we may have been had we not turned. This is my second report of what I feel is a close call in Punta Gorda in the last few weeks. Reaching out to the control facility and making them aware that the pilots are not comfortable with the level of separation being provided might prompt an awareness to the issue. The airport is extremely busy with commercial and training flights along with GA based or visiting the airport. They are constantly having student pilots "making 360s" or slowing to accommodate space for faster or departing traffic. A GA aircraft exiting the runway just the other day blew a tire. He commented to the Tower that he was just trying to clear quickly for everyone. They are constantly having people expedite this or that. They've gotten verbal heated when a GA flight initiated his own go-around because another aircraft was on the runway. They argued about having the required separation on the runway and were upset that he didn't land. The other pilot I was flying with and myself had never heard of such a thing. It is always the pilot's prerogative to decide to go around or not. They are pushing people past their comfort zones and someone is going to get hurt."