I am not a software development or technology expert, nor have I managed this type of a company. However, I am an efficiency and cost control expert and will address some of your concerns.
It is normal that young companies in all fields will encounter efficiency problems after a prolonged time of growth. As a company expands, it must evaluate and re-invent its structures and processes on a regular basis.
The more employees a company has the more are processes and procedures diluted and outgrown. A process evaluation originating from the different tasks and services would give indications of strength and weaknesses.
A further point might be that your department leaders are more technology oriented then drawn to business management. The size of their duties could have outgrown their management experience or capabilities. The implementation of simple but well designed structures with measurable processes will give a good management foundation for these department heads. Business coaching classes or mentor programs will further amplify new found management capabilities.
It seems to me that you know exactly what the development of a new website or an app should cost and what resources must be called upon. You also know how much the customer will have to pay for this service in order to create a sufficient gross margin. I do not doubt that you have metrics and measures in place for these day to day operations. If this is the case and you have healthy margins, your efficiency problems stem from structural, administrative and goal setting deficiencies.
It is most likely that your corporate structural set-up (organization), chain of command and departmental goal setting need an overhaul. Streamlining procedures that mirror present and future needs and that increase ease of execution.
You likely experience also an expenditure and cost overrun problem that needs to be harnessed. Fast growth creates in many cases a cumbersome administration that tries to capture intelligence in a catch-up setting and often tends to cause operational grid lock.
I also expect that the overall corporate culture has changed during the last few years, causing a divergence from the original mission and vision. This cultural change, if not captured, can even derail the most recent corporate goal settings. The more people that work in a place the more cultural change is happening. This does not need to be a bad thing, but the company has to account for it in its most recent business model.

Without knowing details about your enterprise, I would strongly advise that you look for the help of an experienced operational expert. An efficiency strategist that will lead the design and implementation of growth oriented processes and procedures that are coupled with metrics and goal settings. The refined structural or organizational modifications must reflect the values and the vision of the company. This process will involve a task force from your company, people chosen from different areas and rank. The active involvement of this group will ensure that your company values and characteristics are carried in the plan and the company employees have a solid buy-in.
Changes might be subtle or they can be sever in places. Only an analysis of the present situation can produce a conclusive answer.
Please let me know if I can be of further help, either by answering specific questions or by further explaining the process improvement actions.
Please feel free to contact me with your follow up questions - I'll be happy to help.