Exchange rates affect several macroeconomic parameters in open economies. Considering their importance, the USD/TRY and EUR/TRY exchange rates in Türkiye are modelled employing machine learning methods in this study. The exchange rate data are gathered from the official sources for the period of 2003-2023 and then their nonlinearities are inspected in EViews software. As the next step, a machine learning model, namely an autoregressive deep learning model is developed in Python programming language. The developed model is trained separately for the USD/TRY and EUR/TRY exchange rate data. The loss curves regarding the training phases show that the developed model is trained effectively for the exchange rate data. Then, the actual exchange rate data and the results of the developed autoregressive deep learning model are plotted in the same axes showing overlap in a wide range. The performance metrics of the developed model for the USD/TRY and EUR/TRY modelling such as the coefficient of determination, mean absolute error, mean absolute percentage error and the root mean square error are also calculating further verifying the accuracy of the developed autoregressive deep learning model. It is argued that the developed model can also be applied for the modelling of the nonlinear econometric data for other cases.