If it was me, I'd want to be able to fit a growth model myself without a specific package first to make sure that I understand it completely. Then, if there is something specific that a package can do for me, consider using it in addition to my own solution.
I am not an expert on cell growth, but I have done some work in the area. If there isn't a stationary phase or death phase, you might be able to fit a linear model on the log of cell concentration or OD and use the slope and intercept to measure the lag time and first order growth rate. For example from this page of Bioprocess Engineering (Liu) text.
If there is a stationary phase, then you need a more complex model. There is a "logistic" growth model with some parameters you can fit to data with nonlinear optimization (optim in R). The function is too complex in this case to allow a linear regression solution.
If there is a stationary phase and a death phase you want to model, I believe you have to then solve a set of ODEs.